Theory of Substitutions and its Applications to Algebra

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THE

THEORY OF SUBSTITUTIONS A:\111 TT:-
r-ofessor of :r-..1at.he1nat.ios in t.he.

{Jniversit.y of Michinan.

ANN ARBOR, MICH.: 'fHE REGISTER PUBLISHING CO~IPANY-

'CIJC 11nlant> W:,rcss. 1892.

PREFACE. The presentation of the Theory of Substitutions here given differs in several essential features from that which has heretofore been customary. It will accordingly be proper in this place to state in brief the guiding principles adopted in the present work. It is unquestionable that the sphere of application of an Algorithm is extended by eliminating from its fundamental principles and its general structure all matters and suppositions not absolutely essential to its nature, and that through the general character of the objects with which it deals, the possibility of its employment in the most varied directions is secured. That the theory of the construction of groups admits of such a treatment is a guarantee for its far-reaching importance and for its future. If, on the other hand, it is a question of the application of an auxiliary method to a definitely prescribed and limited problem, the ela_boration of the method will also have to take into account only this one purpose. The exclusion of all superfluous elements and the increased usefulness of the method is a sufficient compensation for the lacking, but not defective, generality. A greater efficiency is attained in a smaller sphere of action. The following treatment is calculated solely to introduce in an elementary manner an important auxiliary method for algebraic investigations. By the employment of integral functions from the outset, it is not only possible to give to the Theory of Substitutions, this operating with operations, a concrete and readily comprehended foundation, but also in many cases to simplify the demonstrations, to give the various conceptions which arise a precise form, to define sharply the principal q_uestion, and-what does not appear to be least important-to limit the extent of the work. The two comprehensive treatises on the Theory of Substitutions which have thus far appeared are those of J. A. Serret and of C. Jordan. The fourth section of the "Algebre Superieure" of Serret is devoted to this subject. The radical difference of the methods involved here and there hardly permitted an employment of this highly deserving work for our purposes. Otherwise with the more extensive work of Jordan, the "Traite des substitutions et des equations algebriques." Not only the new fundamental ideas were taken from this book, but it is proper to mention expressly here that many of its proofs and pro,

iv

PREFACE.

cesses of thought also permitted of being satisfactorily employed in the present work in spite of the essential difference of the general treatment. The investigations of Jordan not contaiued in the" Traite" which have been consulted are cited in the appropriate places. But ,vhile many single JJarticulars are traceable to this "Traite" and to these investigations, nevertheless, the author is indebted to his honored teacher, L. Kronecker, for the ideas which lie at the foundation of his entire work. He bas striven to employ to best advantage the benefit which he has derived from the lectures and from the study of the works of this scholarly man, and from the inspiring personal intercourse with him; and he hopes that traces of this influence may appear in many places in his work. One thing be regrets: that the recent important publication of Kronecker, "Grundzuge einer arithmetischen Theorie der algebraischen Grossen," appeared too late for him to derive from it the benefit which he would have wished for himself and his readers. The plan of the present book is as follows: In the first part the leading principles of the theory of substitutions are deduced with constant regard to the theory of the integral functions; the analytical treatment retires almost wholly to the background, being employed only at a late stage in reference to the groups of solvable equations. In the second part, after the establishment of a few fundamental principles, the equations of the second, third and fourth degrees, the Abelian and the Galois equations are discussed as examples. After this follows a chapter devoted to an arithmetical discussion the necessity of which is there explained. Finally the more general, but still elementary questions with regard to solvable equations are examined. STUASBBURG, 1880. To the preceding I have now to add that the present translation differs from the German edition in many important particulars. Many new investigations have been added. Others, formerly included, which have shown themselves to be of inferior importance, have been omitted. Entire chapters have been rearranged and demonstrations simplified. In short, the whole material which has accumulated in the course of time since the first appearance of the book is now turned to account. In conclusion the author desires to express his warmest thanks to ~ir. F. N. Cole who has disinterestedly assumed the task of translation and JJerformed it with care and skill, EUGEN NETTO. GIESSEN, 1892.

TRANSLATOR'S NOTE. The translator has confined himself almost exclusively to the funct!on of rendering the German into respectalJle English. My thanks are especially due to The Register Publishing Company for their generous assumption of the expense of publication and to Mr. C. N. Jones, of Milwaukee, for valuable assistance while the book was passing through the press. F. K. COLE. Ann Arbor, February 27, 1892.

TABLE OF CONTENTS. J>AHT I. THEORY OF SUBSTITUTIONS AND OF INTEGRAL FUNCTIONS.

CIIAI,TER I. SYMMETRIC OR SINGLE-VALUED FUNCTIONS-ALTERNATING AND TWO-VAL-'

~s

UED FUNCTIONS.

1-3.

4.

5-10. 11.

12. 13. 14.

15. 16-20.

Symmetric and single-valued functions. Elementary symmetric functions. Treatment of the symmetric functions. Discriminants. Euler's formula. Two-valued functions; substitutions. Decom1Josit10n of substitutions into transpositions. Alternating functions. Treatment and group of the two-valued functions. CHAPTER IL

MULTIPLE-VALUED FUNCTIONS AND GROUPS OF SUBSTITUTIONS.

22. 24. 25. 26---27. 28. 29-32. 34. 35. 36-38. 39-40.

Notation for substitutions. Their number. Their applications to functions. Products of substitutions. Groups of substitutions. Correlation of function and group. Symmetric group. Alternating group. Construction of simple groups. Group of order pf. CHAPTER III.

THE DIFFERENT VALUES

OF A MULTIPLE-VALUED

FUNCTION AND THEIR

ALGEBRAIC RELA'fION TO ONE ANOTHER.

41-44.

Relation of the order of a group to the number of values of the corresponding function.

viii

CONTENTS.

Groups belonging to the different values of a function. Transformation. The Cauchy-Sylow Theorem. Distribution of the elements in the cycles of a group. Substitutions which belong to all values of a function. Equation for a p-valued function. Discriminants of the functions of a group. Multiple-valued functions, powers of which are singlevalued.

45. 46-47. 48-50. 51.

52. 53. 55. 56--59.

CHAPTER IV. TRANSITIVITY A..."ID PRIMITIVITY.-SIMPLE AND COMPOUND GROUPS.ISOMORPHISM.

Simple transitivity. Multiple transitivity. Primitivity and non-primitivity. X on-primitive groups. Transitive properties of groups. Commutative substitutions; self-conjugate subgroups. Isomorphism. Substitutions which affect all the elements. Limits of transitivity. Transitivity of primitive groups. Quotient groups. Series of composition. Constant character of the factors of composition. Construction of compound groups. The alternating group is simple. Groups of order pa.. J>rincipal series of composition. The factors of composition equal prime numbers. Isomorphism. The degree and order equal. Construction of isomorphic groups.

60-61.

62-63. 64.

65--67. 68. 6!J-71. 72-73. 74-,6. 77-80. 81-85. 86. 87. 88--89. 91.

92. 93. 9i. 95. 96. 97-98. 119--101.

CIIAPTl~R V. ALGEBRAIC

RELATIONS

BETWEEN

FUNCTIONS

BELONGING

TO

THE SAME

GROUP.-FAMILIES OF MULTIPLE-VALUED FUNCTIONS.

103-105. 100.

107.

Functions belonging to the same group can be rationally expressed one in terms of another. Families; conjugate families. Subordinate families.

ix

CONTENTS.

Expression of the principal functions in terms of the subordinate. The resulting equation binomial. Functions of the family with non-vanishing discriminant.

108-109. 110. 111.

CHAPTER VI. THE NUMBER OF THE VALUES OF INTEGRAL FUNCTIONS.

112.

113. 114-115. 116. 117-121. 122-127.

Special cases. Change in the form of the question. }'unctions whose number of values is less than their degree. Intransitive and non-primitive groups. Groups with substitutions of four elements. General theorem of C. Jordan. CHAPTER VII. CERTAIN SPECIAL CLASSES OF GROUPS.

128. 129. 130. 131. 132-135.

136. 137-139.

l'reliminary theorem. Groups n with r = n = p. Cyclical groups. Groups n with r = n =p. q. Groups n with r = n = p2. Groups which leave, at the most, one element unchanged.Metacyclic and semi-metacyclic groups. Linear fractional substitutions. Group of the modular equations. Groups of commutative substitutions. CHAPTER VIII.

ANALYTICAL

140. 141. 143. 144. 145. 146---147.

REPRESENTATION OF SUBSTITUTIONS.-THE

LINEAR

GROUP.

The analytical representation. Condition for the denning function. Arithmetic substitutions. Geometric substitutions. Condition among the constants of a -geometric substitution. Order of the linear group.

PART IL APPLICATION OF THE THEORY OF SUBSTITUTIONS TO THE ALGEBRAIC EQUATIONS.

CHAPTER IX. THE

EQUATIONS OF

THE SECOND, THIRD AND FOURTH DEGREES.-GROUl' OF AN EQUATION.-RESOLVENT8.

148.

The equations of the second degree.

pf!, I ., In accordance with this convention, c 1 , c 2 , c3 , ••• c"-, ... haYe for their highest terms respectively

(7)

c1

= S (x,),

x,,

x, X2 , X1 Xz X3,

•••

x, Xz X3 , . , x .... ,

ct cl c3Y ••• has for its highest term .. ::Ct + ll + Y + .. · xl + Y + .. · X 3Y + ; --. . .

and the £unction

In order, therefore, that the highest terms of the two expressions, Ci"'

cl c3Y ••• and c,a' cl' c3Y' a

•••

may be equal, we must have

+ /1 + r + ... = a' + {J' + r' + ... 11

+r+ r+ ... =

'+r'+ ... r'+ ...

11

that _is, a= a', ;1 = /J', r = r', ... It follows that two different systems of exponents in ct cl c3Y ••• give two different highest terms in the x.\'s. Again it is clear that is the highest term of the expression ci'1 -ll cl- Y c3Y-a ••• and that

* Demonstratio nova altera etc. Gesammelte Werke III,§ 5, pp. 37-38. Cf. Kronecl,er, lionats\Jericllte der Berliner Akademie, lSS!J, p. 913 seq.

6

THEORY OF SlTJlSTITUTIOXS.

all the terms in the expansion of this expression in terms of the x/s are of the same degree. § 6. If now a symmetric function S be given of which the highest term is the difference S- A

C1a-fl

cl-y C3y-B

• • •

= S1

wil! again be a symmetric function; and if, in the subtrahend on the left, the values of the c/\'s given in (7) be substituted, the highest term of S will be removed, and accordingly a reduction will have been effected. If the highest term of S 1 is now A 1 x 1a' ;;e,fl' ;J.?' .1}' ... , then

is again a symmetric function with a still lower highest term. The degrees of S 2 and S 1 are clearly not greater than that of S, and since there is only a finite number of expressions x/· x 2µ x 3v ••• of a given degree which are lower than ;J."1a xl x 3Y • • • , we shall finally arrive by repetition of the same process at the symmetric function O; that is i

S,k -

A

k

C a(k) - 13(k) C fl(k)-y(k) 1

2

• • •

=

O·'

and accordingly we have S

= ~-11 C1a-{3 cl-y .. , + A,c1a'-.6' c,fl'-y'.,. + ...

+

Ak c/k) _fl(k) c/")-y(k) •••

§ 7.

It is also readily shown that the expression of a symmetric function of the x/\'s as a rational function of the c.,,'s can be effected in only one way. For, if an integral symmetric function of x 1 , x,, ...•r,. could be reduced to two essentially different functions of c17 c2 , • • • c,., ,'s (§5). Consequently among these highest terms there must be one higher than the others. But the coefficient of this term is not zero; and consequently(§ 2 (C)) the function '!'-¢ cannot be identically zero. vVe have therefore

Thcoren1 II.

An integral symmetric function of x 1 , x 2 , ••• x,. can always be expressed in one cmcl only one icay as ctn integtal function of the elementary symrnetric functions c1 , c2 , ••• c,..

§ 8. If we write s,._ = S ( x 1"-) for the sum of the ). powers of the n elements x 1, x 2 , ••• xn, we might attempt to calculate the s,._'s as functions of the ci's by the above method. It is however simpler to obtain this result by the aid of two recursion formulas first given by Newton* and known under his name. These formulas are A) s,.-c1s,._ 1+c 2 s,._ 2 B) s,.-c 1s,._ 1 +c2 s,._ 2 -

•••

•••

r > n)

+ (-1 )"c,.s,._,.=0

(

+ ( -l)r 1· c,.= 0

( 1·
-., and taking the sum over).= l, 2, .. . n. The formula B) may be verified with equal ease as follows. If we represent the elementary symmetric functions of ;:r2 , x 3 , • • • x,. by ci', c,', ... c'n-i, we have C1=x1+c.',

C2=X1ci'+c/,

Ca=X1c/+ca', ...

and accordingly, if r < n, we have x t - Ci xt- 1 + c2X 1 r-I_ ... (-l)rc,. = xt-(x1 +ct') xt- 1 (x 2 c/ +c/) xt- 2- ... + (-lY (x1c',._ 1+ c,.'.) = ( -l)Tc,.' and hence, replacing x 1 successively by x,, x 3 , ••• x,., and, correspondingly, c,.' by c,", c,.'", ... c,.("), and taking the sum of the n resulting equations s,.-c 1 s,._ 1 + c2 s,._ 2 - ••• (--l)rc,. n = (-] )r ( c,.' + c,." + + c,.(" ).

+

.....

c:",+ ...

*Newton: Arith. Univ., Ue Trans~tione Aequationum.

8

THEORY OF SUBSTITUTIONS.

The right member is symmetric in x 1 , x 2 , • •• x., and contains all the terms of c,. and no others. :Moreover, the term x 1 x 2 • • • x,., and consequently every term, occurs n - 1 • times. Accordingly we have

s,.-c 1 s,._ 1

+ c2s,._2-

• • , S,. -

C1 S, _ 1

...

+

+ (-1)1' c,.n = (-I)r (n-1·)c,,

C2 S,. _ 2 - - ••• ( -

1 )1" C,.1'

= 0,

and formula B) is proved.* The formula A) can obviously be verified in the same way. § 0. The solution of the equations A) and B) for the successive values of the sll's gives the expressions for these quantities in terms of the c/s. The solution is readily accomplished by the aid of determinants. We add here a few of the results. C) s0 = n s, = c, s2 = c/ - 2c,

t

s3

s,

= c/ = c/ = c/ -

+ 3c + 4c,c + 2c/ - 4c, 5c/r + 5c/c + 5c,c/ - 5c,c, 3ci1· 2

3

4c/c2

3

+

5c,c3 5c5 3 It is to be observed here that all the c;..'s of which the indices are greater than n are to be taken equal to 0. This is obvious if we add to the n elements x,, x 2 , ••• x,. any number of others with the value O; for the ell's up to c" will not be affected by this addition, B;;

while

Cn+I, c.+2, •.•

2

will be 0.

§ 10. The observation of § 5 that ct c/1 c3Y ••• gives for its highest term

1 x«+/3+y+ .. ,x_l 1 2 +y+ .. ,x.Y+~ 3 , , ..,

can beemploved to facili•

tate the calculation of a symmetric function in terms of the c;..'s. \V'e may suppose that the several terms of the given function are of the same type, that is that they arise from a single term among them by interchanges of the x,._'s. The function is then homogeneous; suppose it to be of degree Y, \Ve can then obtain its literal part at once. For, if the function contains one element, and consequently all elements, in the mth and no higher power, then every term of the corresponding expression in the ell's will be of degree m at the highest. For, in the first place, two different terms c,« c/1 c,/ ... and •Another, purely arithmetical, proof ls given by Euler; OJ)uscuh Varii Argument!. Demonstrat. p;enuina theor. Newtonian!, II, p. 108. +Cf. Fail di Bmno: Formes Binaires.

9

SYMl\IETRIC AND TWO-VALUED FUNCTIONS.

er

ct' C3Y give different highest terms in the x/s, so that two such terms cannot cancel each other; and, in the second place, ct c! c3Y ... gives a power x>,'' +fl+ Y + ... , so that 1

•••

a

Again the degree of

+ fi + r + . . . < rn. xt

+ fl +

Y

+ ··· X-:i fl+

Y

+ ·· ·

x3 Y

1s

+

a+2/J+3r+ ... =, and

smce

the given expresssion is homogeneous, the sum must be equal to Y for every term ct c! c3Y • • • These two limitations imposed on the exponents of the c/s that

a+ 2 ;1 + 3 r a+

/3

+ ...

+ r + ... < m,

a+ 2 /1

+ 3 r ... = ,,

€xclude a large number of possible terms. The coefficients of those that remain are then calculated from numerical examples. The 9-uantity a 2 /1 3r is called the weight of the term ct c! c3Y ••• and a function of the c1,.'s, and, being the simplest function with this property, is itself the discriminant. It contains ½n ( n - 1) factors of the form (x>-.-xµ)2; its degree is n ( n-1 ), and the highest power to which any X>-. occurs is the' (n - l)th. It is the square of an integral, bnt, as we shall presently show, unsymmetric function, with which we shall hereafter frequently have to deal. § 12. Finally we will consider another symmetric function in. which the discriminant occurs as a factor. Let the equation of which the roots are Xi, x 2 , ••• x,. be, as before, f (x) = 0. Then if we write

11

SYIUIETRIC AND TWO-VALUED FUNCTIOXS.

clf(x) dx

=

we have, for all values).

f'(x,..)

= (x,.. -

x 1) (x,.. - x 2 )

=

f'(x) ·

1, 2, ... n, the equation (x,.. -- x,.._ 1) (x,.. - x,.. + 1)

• ••

•••

(x,.. - x,.).

\Ve attempt now to express the integral symmetric function S [xi" . f'(x,) . f'(x 3)

•••

f'(x.,)]

in terms of the coefficients c 1 , c2 , ••• c,. of f ( x ), Every one of the n terms of S is divisible by x 1 - x,, since either f '(.r1) or f'(xt) occurs in every term. Consequently, by the same reasoning as in § 11, S is divisible by ( x 1 - x, )2, and therefore being a symmetric function, by every ( x., - x13)2, that is by

(). < 11-;

= 1, 2, ... n -

).

l; :,.

= 2, 3,

...

11 ).

S is therefore divisible by the clisc1·iininant off ( x ), i. e., by the discriminant of the n roots of f ( x ). :N'ow /( x,..) is of degree of n - l in x,.. and of degree 1 in every other x"; and therefore -.J.'1

"./'(x,)./'(r3 )

•••

x 2"./'(x1 ).f'(x3 )

•••

f'(x,.) isofdegree a+n-l in x 1 /'(x,.) isofdegree 2n-3 in x 1 •

Consequently, if a < n - l, S is of degree 2 n -3 in x 1 , while 2 in x 1 • But since .J is a divisor of S, it follows that Sis in this case -identically 0. J is of degree 2 n -

(a< n -1.)

(0)

Again, if a = n - l, then S and .J can only differ by a constant factor. To determine this factor we note that the first term of S is of degree 2 n - 2 in X\, while all the other terms are of lower degree in Xi- 'rhe coefficient of x/"- 2 is therefore

(- l)"- 1 (x2 - x3)

•••

(x2 -

x,.) (x3 -

x,) ... (x3 -

x.) ...

n(n-1)

(xn-:.t·,). • .(x.. -X,._1)=(- l)-2-(x2-X3}"(:.t·2-X,y. • • ( X,,_l -

X,i)2.

In .d the coefficient of

(x2 -

Xs) 2

x/•(x2

2

--

is

x,) 2 •.• (x,._ 1 -

x,.) 2 • n(n-1)

The desired numerical factor is therefore ( -- 1 )-2- and we have

12

THEORY OF SUBSTITUTIOXS. n(n-1)

S[xt- 1 .f'(x1 ).f'(x2 ) ••• f'(x.,)] = (-1)_2_ J. Formulas (9) and ( 10) evidently still hold if we replace

(10)

a\ n - l by any integral function 'f (.r) of degree a

.:S. n

X 1"

or

respectively.

l\Ioreover since we ha,e

(D) according as the degree of 'i is less than or equal to n -

1.

§ 13. If an integral function of the elements x 1 , a·2 , • • • x .. is not symmetric, it will be changed in form, and consequently, if the x;..'s are entirely independent, also in value, by some of the possible interchanges of the x;..'s. The process of effecting such an interchange we shall call a substitution. Any order of arrangement of the .r;..'s we call a permutation. The substitutions are therefore operations; the permutations the result. Any substitution whatever leaves a symmetric function unchanged in form; but there are other functions the form of which can be changed by substitutions. For example, the functions I x/- x/ + x/ - x,2, x 1 x/ x 3 + :X\ x, + x 6 , i:/ + x/ + a·3 take new values if ~ertain substitutions Jje applied to them; thus if x 1 and .r2 be interchanged, these functions become II

-.r/

+a'/+ xa2-x/,

x/x2 x8 + :1ft-X5 + X 6 , x/ + x/ + X 8 •

The first two functions are unchanged if x 1 and x 3 b~ interchanged, the second also if x, and x 5 be interchanged, etc. Functions are designated as one-, two-, three-, m-valued according to the number of different values they take under the operation of all then! possible substitutions. The existence of onevalued functions was apparent at the outset. vVe enquire now as to the possibility of the existence of two-valued functions. In § 11 we have met with the symmetric function J, the discriminant of then quantities x 1 , xt, ... x". The square root of .J is also a rational integral function of these n quantities: *The formula D Is due to Euler; Cale. Int. H § 1169.

SYlIMETRIC AND TWO-VALUED FUNCTIONS.

,vJ =

13

(x 1-x2) (a1\-X3) (x1-x,) ... (x 1 -x,.) (x2 - - x 3 ) (x2 - x,) ... (x 2 - x,.) (x3 - x,) ... (a·3 - x,.)

(x,._ 1 - x,.). Every difference of two elements Xa - xfl occurs once and only once on the right side of this equation. Accordingly if we interchange the xA's in any way, every such difference still occurs once and only once, and the only possible change is that in one or more cases an x" - .-rfl may become xfl - Xa• The result of any substitution is thei·efore either+ ,vJ or - ,vJ, i. e., the function ,vJ is either one-valued or two-valued. But if, in pa1:ticular, we interchange x 1 and .r2 , the first factor of the first row above changes its sign, while the other factors of the first row are converted into the corresponding fact-0rs of the second row, and 't:ice versa. Ko change occurs in the ot,her rows, since these do not contain either x 1 or x 2 • Since then, for this substitution, ,v'J becomes - ,v'J, it appears that we have in ,vJ a tu:o-valued function. This function is specially characterized by the fact that its two values only differ in algebraic sign. Such two- valued functions we shall call alternating functions. Theoren1 III. The square root of the discri,ninant of the n (JUcmtities Xi, x 2 , • • • a·,. is an alternating function of these quantities. § 1-i. Before we can determine all the alternating functions, a short digression will be necessary. An interchange of two elements we shall call a fransposition. The transposition of Xa and Xfl, we will denote by the symbol (::ra xfl). We shall now prove the following

Thcoren1 IV.

Every substitution can be replaced by a

series of tmnspos-itions. Thus, if we have to transform the order x 0 x 2 , J.'3 , • • • x,. into the order x;,, x;0 , X;3 , • • • x;n, we apply first the transposition (xi x;,). The order of the xll's then becomes x;,, x 2, x 3 , •.• x;, - 1 x 1 , x;, + 1 , .•• x,., and we have now only to convert the order x 2 • • • x;, _ 1, X 1 , xi, + 1 , ••• x,. into the order x;2 , Xi3 , • • • xi,.. By

14

THEORY OF SFBSTITUTIONtl.

repeating the same process as before, this can be gradually effected, and the theorem is proveJ. Since a symmetric function is unaltered by any substitution, we obtain as a direct result

Theorcnt Y.

A function ichich is unchanged by ecery transposition is symmeti·ic.

§ 15.

There is_ therefore at least one transposition which ehanges the rnlne of any alternating function into the opposite Yalue. We will denote this transposition by ( .:r.. x·13) , and the altr-rnating function by C:·, and accordingly we have Accordingly, if x .. equation

=

x 13 , we must have C:•

=

0.

Consequently the

regarded as an equation in z has a root z = x 13 and the polynomial ,,,, is therefore cfo-isible by z - Xf!. The function C:• (x1,

;r2, ••.

x .....

X13 •••

xR)

,r.. -

therefore contains x·/l as a factor, and, consequently, c/• 2 contains (x.. - x/l)2 as a factor. But since, for all substitutions, c/• either remains unchanged or only changes its sign, c/• 2 must be a symmetric function; and, accordingly, since c/•2 contains the factor (x .. - x 13)2, it must contain all factors of the form (.:r" - x·,,.) 2, i. e., c/• 2 contains J as a factor, and consequently,:, contains ,vJ as a factor. The remaining factor of c/• is determined by aid of the following

Thcorc1n YI. E,:ery alternating integral function is of the fonn S. ,v j, where ,vJ is the squm·e mot of the discriminant and Sis an hitegml symmeti-ic function. That S. ,vJ is an alternating function is obvious. Conversely, if /3 Tl (x.,-x~Y,

the discriminant of .f(x).

Suppose that Ll' is

the highest power of J which is contained as a factor in J 9 , then, as J contains n(n-1) factms ,l"a-x~, and consequently J' contains n(n-l)t such factors, we must have

. f' [u(n-1) -q ] , n(n-l)t> 2 .

· ·

t > !:.. _ __:l_f'_ = 2 n(n-1)"

'fhe numbe1· t can be O only when q

n(n-1) h . h = -2- - , t at rs, w en all

the transpositions occur in G,. I" is then symmetric and /' = 1. Again q can be O only when G contains no transposition. One of the cases in which this occurs is that where G is the alternating group or one of its subgroups.

Theo rein XII. If ,p is 1,-valued function of the n eleme,ds x 1 , x 2 , • • • x n, the group of n·hich contains q transposif inns, the cliscrimincmt J,p of the f' values of 2. In this case, by anticipating the conclusions of the next Section, we can add the following

89

GENERAL CLASSU'ICATION OF GROUPS,

Coro Ua.ry. If a k-fold transitit•e wou11 k> '!. cmda,ius s!tl.islitntions, different from identity'. il'hich affect 1u/~1ih,-ce than (2k- 2) elements, it is either the altematiug or the S!J11111wtric woup. \Ve may now combine this result with tho corollary of Theoren1 IX. If G is k-fold transitive, it contains suL;ititution-; of the class (n-k+l). Acconlingly q (2k-'!.). Consequently

(n- k + 1) 2:. (2k- 2) and k < n

Thcorein XV.

!

3_

If a gl'Oup of degree n is neitlic1· the alte1·-

nating nor the syminctric m·m1p, it is,

cit the most, ( ; + 1)-folcl tran-

sitfoc.

That the upper limit of transitivity here assigned may actually occur is demonstrated by the five-fold transitive group of twelve elements discovered by Mat\hieu,

G = j (,:rx,x,;r,1) (x4X';;,l'6 X 7), (xx5x,.r,) (x1x,x3:r6), (!/ 1x) (.x·1:r6) (x3 x,) (x,x,), (y2y 1) (:t\;,r3) (x,.r7) (.x·5 x,),

(Y:iY,) (x,.r,) (x3x,) (x,x6), (!J,Y3)

( •.t\X3)

(x,x5) (x 0a·1 ) f.

~ i\). Theoren1 XVI. If a k-fold tran8ifice yroap (k > 1) contain8 a cfrculw· substitution of three ele11tcuts, it colllains file alternating group.

Suppose thats= (x1.r,x3 ) occurs in the given group G. Then, since G i:-i at least, two-fold transitive, it must contain a substitution ,r = (a:J (:1· 1.:t-,:t·>. ... ) ..• aw] consequently also - - -- 's - - (=:J~,,,~JJ. ) ' •

-

o

fJ -

M

.j.•Jl

In the same way it appears that G contains

(:x: 1x,x5), (.r1a·,.rc), ... Cousequcntly (§ 35j G contains the alternating group.

Theorem XVII. If a k-fold t1-ai1sitive u1·otrJJ (k tains a transposition, .the gl'onp is 8,1J1ninetl'ic.

>

I) wn-

The proof is exactly analogous to the preceding. For simply transitive groups the last two theorem--i, where x 1 , x 2 , • • • xk- i may be selected a1·bitr-aril!J. ·we take H 1 =Hand transform H with respect to all the substitutions of G into Hi, H' 1 , H''i, . . . Now let H'i bo that ono of the transformed groups which connocts the k elements x 1 , x 2 , • • • xk of H 1 with other elements, but with the smallest number of these. We maintain that this smallest number is one. For if several new elem on ts .;\, .;2 , • • • occurred in H' i , then from Theorem VIII there must be in the primitive group H'i a substitution which replaces one !; by another !; and at the same time replaces a second !; by one of the elements x 1 , x 2 , • • • xk. Suppose that

is such a substitution, the case whero ,"J = r being included. Then ll''i =fHif- 1 will still contain fy but will not contain ~a- H" 1 , thPJ"efore contains fewer new ek•ments !; than H' 1 • Consequently if H'i is properly chosen, it will contain only ono new element, say Xi-+ 1 It will thereforo not cont,ain some one of the elements of H 1 , say x a• W o select then from H 1 a Fmbstitution 7t = (. .. x a•T"i. ... ) . . . and form the group u- 1 H'i u= II-;,. This group contains

but not x,... In the same way we can form a group H" which affects only x 1 , J"2 ••• x,.._ 1 , a:i-+ 2 , and so on. It remains to be shown that a· 1 , x 2 , • • • xk- 1 can be taken arbitrarily, that i", that the assumption H = H 1 is always allowabh'. Suppose that H 1 contains x 1 , ai 2 , ••• x,.._,,,. Th0n in the series 11 1 , H,,, ... thero is a group H 0 which also contains JI,.._a+ 1 • Proceodiug from .H0 and the elements x 1 , x 2 , . . . xk-a+i, we construct a sories of groups, as beforo, arriving finally at tho group .II. ~ 83. Theormn XX. If n, 1m:mitii,e grouJJ G of degree n contains a primitive subgroup .II of degree k then G is at least ( n - k 1)-fold transitive.

+

From the preceding thoorem H 1 affects the elements Xi, x~, ... x,..; jJ-f 1 ,H 2 \ the olom. is a maximal self-conjugate subgroup of the preceding one, then this series is called the sm·ies belonging to the compound g1·oup G, or the series of composition of G, or, still more briefly, the series of G. If the numbers

are the orders of the successive groups of tho se1·ies of composition of G, then e,, e2 , ••• e"+ 1 are called the factors of composition of G; and we have r=e 1 e2 e3 • • • Cµ+i·

If, in accordance with the notation of § 86, we write,

+

the order and the degree of every 1'a is equal to ea (fL = l, 2, ... 11. 1). All the groups r .. are simple. For r.. is (l-1·..)-fold isomorphic with Ga-" and to the identical substitution in l'.. corrnsponds G .. in a.. _,. Consequently, if r .. contains a self-conjugate subgroup different from identity, then the corresponding self-conjugate subgroup of Ga_, (§ 73) contains and is greater than G... The latter would therefore not be a maximal self-conjugate subgroup of Ga-I" The groups l', which define the transition from every u.. to the following one m the series of composition, are called the factor groups of G.*

* 0. Holder;

Matb. Ann. XXXIV, !l- 30 ff.

97

GENERAL CLASSIFICATION OF GROUPS.

§ 88. Given a compound group G, it is quite possible that the corresponding series of composition is not fully determinate. It is conceivable that, if a series of composition G, G1 , G2 ,

•••

Gµ,l

has been found to exist, there may also be a second series

G, G'i, G' 2 ,

•••

G'v,1

in which every G' is contained as a maximal self-conjugate subgroup in the preceding one. ·we shall find howeyer that, in. whatever way the series of composition may be chosen, the number of groups G is constant, and moreover the factors of composition are always the same, apart from their order of succession. Suppose the substitutions of G1 and G'i to be denoted by s,,_ and s',,_ respectively. Let r 1 =r :e1 be the order of G1 , and r' 1 =1· :e'i that of G'i. The substitutions common to G1 and G'i form a group I'(§ 4-!), the order x of which is a factor of both r 1 and r'i. ,ve write The substitutions of 1' we denote by a.,. All the substitutions of G 1 may then be arranged in a table, the first line of. which consists of the substitutions a,,_ of r We obtain IT 1

.t";

r,

52a2, 521T:J, .•. 52 6 xi

52 1~

5y"I' 5yG2, SyfTs, ... Sl1x;

5yI',

= 1, 52

6 1'

fi2,

l'Tg,

(j

where the 5 belonging to any line is any substitution of G1 not contained in the preceding lines. The group G'i can be treated in the same way. We will suppose that in this case, in place of 51 , 52 , ••• , we have s'i, 5'2 , . . . Every substitution of G' or G' 1 can, then, bewritten in the form s.,= 5/3"-y, s'.,= 51f3"r Again, the product Sa.

-1 f

S f3

-1

I

B.,B f3

=

r -1( I

B o.

S f3

-I

I)

S.,S f3

=

(

Sa.

-1 f

S f3

-1

)

I

Ba. S f3

belongs to G1 • For, since G- 1 G1 G= Gn it follows that s'13 - 1s,,_s'13 , which occurs in the second form of the product, is equal to By, and the product itself is equal to s.,- 1s-y. But, from the third form, this·

98

THEORY OF SUBSTITUTIONS.

same product belongs to G', since G- 1 G'G= G', and therefore sa- 1s'~- 1sa=s'y, so that the product is equal to s'ys'~- Com,equently the product belongs to the group r which is common to G 1 and G'i. Hence In particular, since the 11's belong to both the s's and the s''s, we obtain

B) From this it follows that the substitutions of the f9rm Sas' ~,;Y form a group ill. For, by repeated applicati_ons of the equation B), we obtain (sas'~O'y) (Sa£,' b,;,.) = Sa';;' ~Sa6aS\11c = SaSa • s' ~,;.s\,;, = SaSaS' ~s' b(J'd = '3~11.s' Y/(jf(J'a s~s' Y/11,'t.

=

The group ill is commutative with G; for we have

a-1(s~s' Y/,;-r,)G = G • 1 s~G. a- 1s\r.,'tG = sa.s' ~ = sy11a. s',,;, =sys',,,.•. The group ill is more extensive than G1 or G'; it is contained in G; consequently, from the assumption as to G 1 and G', ill must be identical with G. \ The order of ill is equal to xy -y'. For, if Si,'~r,Y = Sas' b~ci it is .easily seen that a= a, b = (1, c = r. Consequently the order of G is also xyy', and since we have it follows that This last result gives us for the order of J', x - - - , - , = ~. e1e I e I e1 We can show, further, that I' is a maximal self-conjugate subgroup of G1 and of G' 1 , and consequently occurs in one of t4e series of .composition of either of these groups. For in the first place J', as a part of G' 1 , is commutative with G 1 , and, as a part of G1 , is commutative with G' 1 , so that we have -

G1-ll'G1 = G'i,

1·

-

1'1

1·'1

0'1- 11'G'i = G1.

But since the left member of the first equation belongs entiwly to G1 , the same is true for the right membe.r, and a similar result holds _ror the second equation. Consequently

GENERAL CLASSIFICATION OF GROUPS.

99

A.gain there is no self-conjugate subgroup of G 1 intermediate between G1 and I' which contains the latter. For if there were such a group H with substitutions ta, then it would follow from A) that

that is, H is also commutative with G'.. A.nd since G1 and G'. together generate G, it appears that H must be commutative with G. If now we add to the fa's the s12 , s' 3 , • • • , then the substitutions s' atfl form a group. For since 1' is contained in H and in Gu we have from A) (S'af(,) (S'yfa)

= 51

a ·

51../(,a, - ta= 51atb•

This group is commutative with G, since this is true of its component groups H and G' 1 • It contains G' 1 , which consists of the substitutions s' atTfJ • It is contained in G, which consists of the substitutions s' aSfJr,r But this is contrary to the assumption that G'i is a maximal self-conjugate subgroup of G. We have therefore the following preliminary result: If in two series of composition of the group G, the gmups next succeeding Gare respectively G1 and G11 , then in both series we may take for the group next succeeding G1 or G' 1 one and the same maximal self-conJngate subgroup l', which is composed of all the substitutions common to G1 and G'.. If e1 and e'. are the factors of composition belonging to G1 and G11 respectively, then l' has for its f actars of composition, in the first series e' 1 , in the second e1 •

§ 89.

We can now easily obtain the final result.

Let one series of composition for G be 1)

and let a second series be 2)

G, G\, G\, G's, ... ,

Then from the result just obtained, we can construct two more series belonging to G:

100

THEORY OF SUllSTITUTIONS.

3) G, Gi, 1~ .J, H, . . .

4) G, G'i, 1', .J, H, ... ,

and apply the same proof for the constancy of the factors of composition to the series 1) and 3), and again 2) and 4), as was employed above in the case of the series 1) and 2). The series 3) and 4) have obviously the same factors of composition. The problem is now reduced, for while the series 1) and.2) agree only in their first terms, the series 1) and 3), and again 2) and 4), agree to two terms each. The proof can then be carried another step by constructing from 1) and 2) as before two new series, both of which now begin with G, G1 :

1')

G, GI' G2, (B, ~. ~ •... ' 1·, r 1 , r 2 =1\:e2, r" 3 =r2:e'2, ...

3')

G, G1, 1', @, ~' '.j, . - · , 1·, r1, 1·'2=1\:e'2, r".=1·'2:e2,, ..

These series have again the same factors of composition, and l') and 1) and again 3') and 3) agree to three terms, and so on. We have then finally

Theo rein XXIII.

If a compound group G admits of two dijferent se1·ies of composition, the factors of .composition in the two cases are identical, apart from their m·der, and the number of groups in the two ser·ies is therefore the same.

§ 90. From § 88 we deduce another result. Since a- 1rG belongs to Gi, because a-•a1 G = Gi, and also to G11 because a- 1 G'iG=G'i, it appears that, a- 1rG, as a common subgroup of G 1 and G'u must be identical with /~ so that /' is a self-conjugate subgroup of G. From § 86 it follows that it is possible to con-

struct a group

.Q

of order e1e11 which is (1-~ )-fold isomorphic e1C I

with G, in such a way that the same substitution of !! corresponds to all the substitutions of G which only differ in a factor (1. We will take now, to correspond to the substitutions 1, s,. 53 , • • • S/ 1 of G 1 , the substitutions 1, w2 , w3 , • • • w.,, of !!, and, to correspond to the 1, s\, s'3 , • • • s',1 of G'i, the substitutions 1, w\, w\, ; .. ul,. of !2. In no case is s' a= s13sy, for the (l's form the common subgroup of G,

101

GENERAL CLASSIFICATION OF GROUPS.

and G'i. Consequently the w's are different from the w"s. classes of substitutions give rise to groups: Q1

= [l, w

2,

w 3 , •••

w,J,

.Q1

Both

= [1, w\, w\, ... w',J,

and, since sas'll=s'/lsa"-r• it follows that Q 1 Q' 1 =.Q\.Q 1 • every s in G is equal to Sas' /ltYy, that is Q = Q1.Q 11 •

:Moreover

We obtain .Q therefore, by multiplying every substitution of by every one of .Q' 1 •

n1

§ 91. We consider now two successive groups of a series of composition, or, what is the same thing, a group G and one of its maximal self-conjugate subgroups H. Suppose that s'. is a substitution of G which does not occur in H, and let s' i"' be the lowest power of s11 which does occur in H (mis either the order of s' 1 or a factor of the order). If m is a composite number and equal to pq, we put s'. 2 = s1 , and obtain thus a substitution s1 which does not occur in H, and of which a prime power st is the first to occur in H. We then transform s 1 with re!5pect to all the substitutions of G, and obtain in this way a series of substitutions s 1 , s2 , ••• BA. No one of these can occur in H. For if this were the case with Ba= , , - 1s 1a, then GSar.-• s 1 , being the transformed of a substitution Sa of H with respect to a substitution G- 1 of G, would also occur in H. ,ve consider then the group

=

T= jH, s 11 s21 ••• sA} .• This group contains H and is contained in G. substitution of G, we have

If t is any arbitrary

t- 1 J't=t- 1 jHstsl . .. }t=t- 1Ht-t-•sa.t.t- 1slt ... == H Si/St./ . .. == 1: I' is therefore commutative with G. These three properties of I' are inconsistent with the assumption that H is a maximal self-conjugate subgroup of G, unless J' and H are identical. If we remember further that all substitutions, as s,, s2, ... s,., which are obtained fr.om one another by transformation, are similar, we have Thcorcin XXIV. Every g1·oup of the se1·ies of composition of any •grnup G, is obtainable from the next following (01·, every group is obtainable from any one of its maximal self-confugate

102

THEORY OF SUBSTITUTION!:!.

snbgroups) by the addition of a series of substitutions, 1) which are sirnilar to one another, and 2) a prime pou:er of which belongs to the smaller gmup. The last actual gmup of a series of composition consists entirely of similar substitutions of prime order.

§ 02. The following theorem is of great importance for the theory of equations:

Theorein XXV. The series of composition of the symmetric group of n elements, consists, if n > 4, of the alternating group and the identical substitution. The corJ"esponding factors of composition are therefore 2 and ½n! The alternating group of more than four elements is simple.

,ve have

already seen that the alternating group is a maximal self-conjugate subgroup of the symmetric group. It only remains to be shown that, for n > 4, the alternating group is simple. The proof is perfectly analogous to that of § 52, and the theorem there obtained, when expressed in the nomenclature of the present Chapter, becomes: a group which is commutative with the symmetric group is, for n > 4, either the alternating group or the identical substitution. It will be necessary therefore to give only a brief sketch of the proof. Suppose that H 1 is a maximal self-conjugate subgroup of the alternating group H, and consider the substitutions of H 1 which affect the smallest number of elements. All the cycles of any one of these substitutions must contain the same number of elements (§ 52). The substitutions cannot contain more than three elements in any cycle. For if H contains the substitution 8

= (;,:\X~3X4 ... ) •.. ,

and if we transform s with respect to ,; = (x 2 x 3 x,), which of course occurs in H, then s- 1a- 1s11 contains fewer elements than s. Again the substitutions of H 1 with the least number of the elements cannot contain more than one cycle. For if either Ba.= (x1X2) (x3X4). • . , BfJ = (X1X2X3) (x,x.x6) • , •, occurs in H, and if we transform with respect to a= (x 1x 2x 5), the products

will contain fewer elements than

Ba., sfl

respectively.

GENERAL CLASSIFICATION OF GROUPS.

103

The substitutions which affect the smallest number of elements are therefore of one or the other of the forms

Sa= (Xo.Xfl),

t = (XaX13Xy).

The first case is impossible, since the alternating group cannot contain a transformation. The second case leads to the alternating group itself. If n = 4, we obtain the following series of composition: 1) the symmetric group; 2) the alternating group; 3) G2 = [1, (x1x 2) (x3x,), (xix3) (x2x,), (xix,) (x2x3)]; 3) G3= [1, (xiX2) (x3x,)]; 5) G,= 1. The exceptional group G2 is already familiar to us. § 93.

We may add here the following theorems:

Theore1n XXVI. Every g1·oitp G, which is not contained in the alternating group is compound. One of its factors of composition is 2. The corresponding factor gmup is [(l, Zif z2 , ) ] . The proof is based on § 35, Theorem VIII. The substitutions of G which belong to the alternating group form the first self-conjugate subgroup of G.

Tbeorein XXVII. If a group G is of order pa, p being a prime number, the factoi·s of composition of G are all equal top. The group K of order pf obtained in§ 30 is obviously, from the method of its construction, compound. It contains a self-conjugate subgroup L of order pf- 1 and this again contains a self-conjugate subgroup Jl,J of order pf- 2, and so on. The series of composition of K consists therefore of the groups

K,

L, M,

. . . Q, R, ... S, 1,

of orders

pf,pf-l,pf-2, ... p\pk-1,. • .p, 1. The last corollary of § 49 shows that we need prove the present theorem only for the subgroups of K. If G occurs among these and is one of the series above, the proof is already complete. If G does not occur in this series, suppose that R is the first group of the s~ries which does not contain G, while G is a subgroup of Q. We apply then to G the second proposition of § 71. Suppose that H is the common subgroup of R and G. Then H is a self-conjugate subgroup of G, and its order is a multiple of po.- 1 and is conse-

104

THEORY - 01., 7'' >., .•. , and with 7>. all the ,r>., rr' >-, rr" >-, •.. , and proceed in the same way with all the substitutions s>. of G. The rr/s form a group ~: and the 7>.'s a group T. Suppose that ,;>-, ,;,,_ are coordinated with -::>., r,,_. Then there are substitutions S>., s,,_, s,, such that 8>..

== ti;..TA, 8µ. == fiµ.7'/J.' == Bv = ffv7v,

8>..Sµ.

and consequently,;>.,;,,_="• is coordinated with '>.7,,_ = r,.

CHAPTER V. ALGEBRAlC RELATIONS BETWEEN FFNCTlOXS BELONCHNG TO THE SAME GROFl'. FAMILIES OF MULTil'LEYALnm Fl'NCTIONS.

§ 102. It has been shown that to every multiple-valued function there belongs a group composed of all those substitutions and only those which leave the value of the given function unchanged. Conversely, we have seen that to every group there correspond an infinite number of functions. The question now to be considered is whether the property of belonging to the same group is a fundamentally important relation among functions; in particular, whether this property implies corresponding algebraic relations. An instance in point is that of the discriminant J q, of the values of a function c;, considered in Chapter III, § 55. It was there shown simply from the consideration of the group belonging to S", that .1¢, and therefore the corresponding discriminant of any function belonging to the same group, is divisible by a certain power of the discriminant of the elements x 1 , Xi, .•. x •.

§ 103. We shall prove now another mutual relation of great "importance.

Theore1n I. T1co .function.~ belonging to the same u1·oup caii be 1·ationa.lly expressed nne in ter1ns of the othe1'. Suppose c; 1 and 1.\ to be two functions belonging to the same group of order 1· and degree n

If ,r, is any substitution not belonging to G1 , and if -1, it is greater than n. For such a fnnction then, if n > 4, f' cannot be less than n. Again for the primitive groups it follows from Chapter IV, Theorem XVIII, in combination with the first result of Theorem I, § 113, that if p < n, the corresponding group is either alternating or symmetric, that is, p = 2 or 1. The non-primitive group for which n = 4, p = 4, 1· = 8 is already known to us, (§ 46). We have then

Theorent II. If the numbel' /' of the values of a function is less than n, then either p = l or p = 2, and the group of the function is either symmetric or alternating. An exception occurs only for n = 4, p = 3, 1· = 8, the corresponding gmup being that belonging to X1X2 X3X4 •

+

132

THEORY OF SUBSTITUTIONS.

§ 115. On account of the importance of the last theorem we add another proof based on different grounds. Suppose

9, at least 4-fold transitive. G contains therefore the substitutions

, = (x

1)

(x2x 3) (x,x5

••• ) ,

,- 11I1T = (x1Xa) (X2X5), so that we return in every case to the type A). therefore no group of the required type.

For n > 9 there is

Theoren1 III. If the degl'ee of a gl'oup, which contains substitutions of four, but none of three or of two elements, exceeds 8, the group is either intransitive or non-primitive. Combining this result with those of § 113 and § 116, we have

Theorein IV. If the number t> of the values of a function is 1wt greater than ½n(n-l), then (f n > 8, either 1) p = ½n(n-l), and the .fnnction is symmeh·ic in n-2 elenients on the one hand and in the t-wo remaining elenients on the other, or 2) p = 2n, and the function is alternating in n-1 elements, or 3) p = n, and the function is symmetric in n -1 elements, or 4) p = 1 or 2, and the function is symnietric or alternating in all the n elements.* § 122_

We insert here a lemma which we shall need in the proof of a more general theorem. From § 83, Corollary II, a primitive group, which does not include the alternating group, cannot contain a circular substitution

t

-cauchy:Journ. de !'Ecole Polytech. X Cahier; Bertrand: Ibid. XXX Cahier; Abel: Oeuvres completes I, pp.13-21; J. A. Serret: Journ. de l'Ec1>le PoJytecb. XXXII Callier; C. Jordan: Traite etc., pp. 67-75. tC.Jordan: Traiteetc.,p.664. Note·c.

THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS.

of a prime degree less than 23n.

t, and if

than 2

139

If p is any prime number less

pf is the highest power of p which is contained in

n!, then the order of a primitive group G is not divisible by pf. For otherwise G would contain a 15ubgroup which would be similar to the group K of degree n and order pf (§ 39). But the latter group by construction contains a circular substitution of degree p, nl and the same must therefore be true of G. Consequently p = -

r

must contain the factor p ·at least once. What has been- pro..-en for pis true of any prime number less than 2; 1 and consequently for their product.

We have then

Theore1n V. If the group of a function with more than two values is primitive, the number of values of the function is a multiple of the product of all the prime numbers which are wss 2n than 8 . § 123.

By the aid of this result we can prove the following

Theorem VI. If k is any constant number, a function of n elements which is Etymmetric or alternating with respect to n-k of them has fewer values than those functions which have not this property. For small values of n exceptions occur, but if n exceeds a certain limit dependent on k, the theorem is rigidly true.* If ,p is an alternating function with respect to n - k elements, the order of the corresponding group is a multiple of ½(n-k)!, and the number of values of the function is therefore at the most

A)

2n(n-1) (n-2) ... (n-k+l).

If cf1 is a function which is neither symmetric nor alternating in n-k elements, it may be transitive with respect to n-k or more elements. But in the last case c/1 must not be symmetric or alternating in the transitively connected elements.

We proceed to determine for both cases a minimum number of

•c.

Jordan:

Traite etc., p. 67.

140

THEORY OF SUBSTITUTIONS.

values of n-k i.e., n> 2k+l. The maximum order of the group is consequently

+

(n-k-1)! (k+l!, and the minimum number of values of ¢ is 1

B)

n! _n(n-l)(n-2) ... (n-k) ~(1-1-k~l~)!~(-k~+lt) 1-2-3 .... (k+l)

It appears at once that the minimum B) exceeds the maximum A), as soon as

This is therefore the limit above which, in the first case, the theoorem admits of no exception.

§ 125. In the second case Y' is transitive in n-x elements (x > k), but it is neither alternating nor symmetric in these elements. The group G of ¢• is intransitive, and its substitutions are therefore products each of two others, of which the one set o-i, rT2 , • • • connect transitively only the elements x 1 , x 2 , ••• x,. __ ., while the other set r 1 , r 2 , ••• connect only the remaining element& X11-K+t,. • • Xn•

The substitutions of the group G of 4x!2 -! This is shown at once, if we write the right hand member in the form (4l(![k-x]!) ([ -n-r.] - [n-)( - - - 1 ] ... [k-x+l] ·) .

2

2

For the first factor is constant as n increases, and the ratio of the left hand member to the second parenthesis has for its limit

n+K

2-2--i

§ 127. Finally, if the function¢' of the n-l( elements is primitive, we recur to the lemma of § 122. From this it follows that the minimum number of values of (/ is the product of all the prime numbers less than i· (n-Y.). We will denote this product by 1

P[2(n-x)] 3

•

Introducing it in 0), we have p [2(n-r.)] n(n-1) ... (n-Y.

3

+ 1)

x!

We have then to show that, for sufficiently large values of n, the value A) is less than C2), i. e., that P [ 2(n-r.)] 3

> 2x!(n-x) (n-x-1) ... (n-k+l.)

The right hand member of this inequality will be greatly increased if we replace every n-Y.-a by the first factor n-r.. There are k-Y. factors of the form n-Y.-,-a. These will be replaced by

THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS,

(n-xY,-".

.

If we write then v =

2(n-x) 3

143

, we have only to prove

that for eufficiently large v, P(v)

> [2(1Y-• x!]vk-•,

or

This can be shown inductively by actual calculation, or by the employment of the theorem of Tchebichef, that if Y > 3, there is always a prime n·umber between v and 2 Y - 2. For we have from this theorem

P(2v) > vP(Y), (v)k-,c = 2k-,c yk-• P(2v) (2,Y,-•

>

P(v) _,_ VTo-,c 2k-•"

Now whatever value the first quotient on the right may have, we can. always take t so great that the left hand member of

P(2',) (2'vy-,c

P(,) (

> Yk-•

, )' 2k-•

increases without limit, if only Y is taken greater than 2k- •. The proof of the theorem is now complete. The limits here obtained are obviomily far too high. In every special case it is possible to diminish them. As we have, however, already treated the special cases as far as p = ½n(n-1), it does not seem necessary, from the present point of view, to cany these investigations further.

CHAPTER VII. CERTAIN SPECIAL CLASSES OP GROUPS.

§ 128. We recur now to the results obtained in § 48, and deduce from these certain further important conclusions.* Suppose that a group G is of order 1· = p"m, where p is a prime number and m is prime top. We have seen that G contains a subgroup Hof order p". Let J be the greatest subgroup of G which is commutative with H. J contains H, and the order of J is therefore p"i, where i is a divisor of m and is consequently prime to p. Excepting the substitutions of H, J contains no substitution of an order pfJ. For if such a substitution were present, its powers would form a group L of order pfJ. But if in A) of § 48 we take for G1 , H 1 , K 1 thepresentgroups J, L, K, then since ,rY•- 1H,ry=H, we should have p"i pfJ pfJ - = - d +-d

P"

I

2

+ ...

The left member of this equation is not divisible by p. Consequently we must have in at least one case dy = pfJ, that is, L is contained in ,ry- 1H,ry=II. Again, every subgroup M of order pa which occurs in G is obtained by transformation of H. For if we replace GI, HI, K1 of A) § 48 by G, H, M, we obtain p"i pa pa -=-d-+-l pa I ( e

+ ... ,

and for the same reason as before dy = pa in at least one case, and therefore JJI = ,rY - 1Ifoy. Since H, as a self-conjugate subgroup of J, is transformed into itself by the pai substitutions of J, it follows that there are always exactly p"i substitutions of G which transform H into any one of its conjugates. * I,. Sylow:

Math. Ann. V ..584-94.

There are therefore

1 ~ i

of the latter.

CERTAIN SPECIAL CLASSES OF GROUPS.

145

Finally, if we replace G, H 1 , K 1 of A) § 48 by G, J, H, we have

Since H1 is contained in J 1 , we must have d 1 = p"-, and since J contains no other substitutions of order pf\ no other d can be equal to p"-. It follows that r = p"-i(kp+ 1),

rn = i(kp

+ 1).

The group H has therefore kp + 1 conjugates with respect to G. \\re have then the following

Theorent I. If the order r of a group G is divisible by p"but by no highei· power of the prime nimiber p, and if His one of the subgroups of orde1· p"- contained in G, and J of order p"-i the l,argest snbg1·oup of G which is commutatfre with H, then the order of G i43 r = p"-i(kp 1).

+

Every snbyrmip of orda p"- contained in G is conjugate to H. Of these conjugate groups there are kp 1, and everiJ one of them can be obtained from H lJy p"-i different ti-ansfurmaf£ons.

+

§ li9. In the discussion of isomorphism we have met with transitive groups whose degree and order are equal. In the following Sections we shall designate such groups as the groups !!. If we regard all simply isomorphic transitive groups, for which therefore the orders r are all equal, as forming a class, then every such class contains one and only one type of a group !! (§ 08). The construction of all the groups !2 of degree and order 1· therefore furnishes representatives of all the classes belonging to 1·, together with the number of these classes. The construction of these typical groups is of especial importance, because isomorphic groups have the same factors of composition, and the latter play an important part in the algebraic solution of equations.

One type can be established at,once, in its full generality. This type is formed by the powers of a circular substitution. A group !! of this type is called a ,·yct-ical group, and every function of n elements which belongs to a cyclical group is called a cyclical function.

146

THEORY OF SUBSTITUTIONS.

,ve limit ourselves to the consideration of cyclical groups of prime degree p. If s = (xi x 2 . .. xp), and if w is any primitive p th root of unity, then 0. These subgroups would have only the identical substitution in common. They would therefore contain in all

(p-1) (px+ 1)+1 =p[(p-l)x +1]2:pq substitutions.

This being impossible, we must have

x

= 0.

CERTAIN SPECIAL CLASSES OF GROUPS.

147

The subgroup H contains only p substitutions; the rest are all of order q. Their number is pq-p= (q-l)p.

There are therefore p subgroups of order q, and consequently from Theorem I we must have p=).q+ 1,

p-1

).=--, q

that is, q must be a divisor of p-1. be any new type !!.

Only in this case can there

3) The group H is a self-conjugate subgroup of !t Consequently every substitution t of order q must transform the substitutions of H into s", where a might also be equal to 1. We write

s = (x/x/ ... xP') (x/x/ . .. x/) ... (x,'lx/ . .. xP,z), (where the upper indices are merely indices, not exponents). Then no cycle of t can contain two elements with the same upper index. For otherwise in some power of t one of these elements would follow the other, and if this power of t were multiplied by a proper power of s, one of the elements would be removed, ,vith a proper choice of notation, we may therefore take for one cycle of t

It follows then from t-'st

that t replaces x/ by Xa+/+', Xan+/+ 1, ••• so that we have

= sn X3b by X2a+/+1, • .. Xa+i" by

t = (x/x/ ... x/) ... (Xa+/Xaa+/Xan•+/ .•• Xaa,z-t+1q ••• ) ••• If now the latter cycle is to close exactly wiih the element Xaa,z-1+ i'z, we must have aa'l+l-a+ 1, a'l=l (mod. p). The solution a= 1 is to be rejected, for in this case we should have

t = (x/xi' . .. x 1q) (x.}x/ . .. x 2'l) ... (x/xp2 . .. x/), st= (xi'x/ . .. x/X,z+/x2+22 ... ) ... ,

so that the latter substitution would contain a cycle of more than q elements, without being a power of s.

148

THEUh ll OF SUBSTITUTIONS.

It follows then from thil congruence aq==l (mod. p) that q is a divisor of p-1, as we have already shown; further that a, belonging to the exponent q, has q-1 values a,, a 2 , • • • aq_,; finally that all these values are congruent (mod. p) to the powers of any one among them. From t- 'st= s" follows

t- 2 st' = s«•, t-"st" = s«', ..• so that, if s is transformed by t into any one of the powers s«>-, there are also substitutions in Q which transform s into 8°1, s"•, ... s"q-,. Accordingly the particular choice of a>- has no influence on the resulting group, so that if there is any type Q generated by substitutions s and t, there is only one. The group formed by the powers of t being commutative with that formed by the powers of s, the combination of these two substitutions gives rise to a group exactly of order pq. The remaining pq-p-q 1 substitutions of the group are the first q -1 powers of the p-1 substitutions conjugate to t

+

. •)~, 3' .. . p). . (,1= If p and q are unequal, 1ce have therefm·e only one new typp, U.

§ 131. and order p

Finally we determine all types of groups f! of degree 2•

1) The cyclical type, characterized by the presence of a substitution. of order 1l, is already known.

2) If there are other types, none of them can contain a substitution of order p 2• There are therefore in every case p 2 - l substitutions of order p and one of order 1. If s is any substitution of [!, and t any other, not a power of s, then f! is fully determined by a and t. For all the products

8at''

(a, b =0, 1. 2, ... p-1)

are different, and therefore £!

= [s"t1']

(a,

b

= 0, 1, 2, ... JJ-:--1 ).

,ve must have therefore

If now two of the exponents J are equal, it follows from

149

CERTAIN SPECIAL CLASSES OF GROUPS.

that

Ws)-'Ws)

= s-'trs = (s 8 t•)-'(s8 t•') = fY.

Since for tc we may write t, it therefore appears that Q contains a substitution t which is transformed by s into one of its powers v,. The same result holds, if all the exponents 8 are different. For one of them is then equal to 1, since none of them can be 0, and from tas = st• follows s- 1t"s t•.

=

3)

There is therefore always a substitution

t = (xi'x/ ... xP') (x/x/ ... x,,2)

•••

(xtxl ... xPP)

which is transformed by s into a power of itself ceding Section, we may take for one cycle of s

f"'1.

As in the pre

(xi'x/ ... xt). Then from s - 'ts = t" follows

If the second cycle is to close after exactly p elements, we must have aP 1==2, aP==l (mod p).

+

This is possible only if a = 1.

Accordingly

s = (xi'x/ ... xt) (x/x/ ... xl) ... (x/xp 2 • • • / ) . The p

+ 1 substitutions' s, t, st, st2,

•••

stP- 1

are all different and no one of them is a power of any other one. Their first p-1 powers together with the identical substitution form the group Q. Summarizing the preceding results we have

Thcormn II. There m·e three types of groups Q, for which the degree and order are equal to the product of two prime numbers: 1) The cyclical type, 2) one type of order pq (p > q), 3) one type of order p 2• The first and third types are always present; the second occurs only when q is a divisor of p-1. § 132. We consider now another category of groups, characterized by the property that their substitutions leave no element, or

150

THEORY OF SUBSTITUTION:,!.

only one element, or all the elements unchanged. the groups we assume to be a prime number p.

The degree of

Every substitution of such a group is regular, i. e., is composed of equal cycles. For otherwise in a proper power of the substitution, different from the identity, two or more of the elements would be removed. The substitutions which affect all the elements are cyclical, for p is a prime number. From this it follows that the groups, are transitive, and again, from Theorem IX, Chapter IV, that the number of substitutions which affect all the elements is p-1. We may therefore assume that s

= (x x x 1

2

3 • ••

xP)

and its first p-1 powers are the only substitutions of p elements which occur in the required group. The problem then reduces to the determination of those substitutions which affect exactly p-1 elements. If t is any one of these, then t- 1st, being similar to s, and therefore affecting all the elements, must be a power of s

where every index is to be replaced by its least positive remainder (mod p). Since it is merely a matter of notation which element is not affected by t, we may assume that x 1 is the unaffected element. It follows that

If now g is a primitive root (mod. p), then all the remainders (mod. p) of the first p-1 powers of g

G) are different, and we may therefore put

m=g"

(mod. p).

We will denote the corresponding t by fw p-l

consists of µ cycles of - - elements each. µ

closes as soon as

It appears then that

fµ.

For every cycle of

tµ.

151

CERTAIN SPECIAL CJ,ASSES OF GROUPS.

am•+1--a+1, m• __ gµ.•-1,

(mod. p)

p-1

and this first happens when z = - - . fl.

If there is any further substitution t. which leaves x, unchanged and which replaces every Xa+i by Xa 9• +u then tµ.atJJ replaces every Xa+i by Xa 9aµ.+/!•+i• If now we take a and fi so that aµ+fiv is

congruent (mod. p) to the smallest common divisor we have in

w

of 11. and v,

= tµ.o.tJJ

lw

a substitution of the group, of which both lµ. and t. are powers. Proceeding in this way, we can express all the substitutions which leave x, unchanged as powers of a single one among them fu, where gu is the lowest power of g to which a substitution t of the group corresponds. The group is determined by s and tu.

Since tu is of order p - 1 , O"

it follows from Theorem II, Chapter IV, that the group contains in all p(p- l) substitutions.

IT

may be taken arbitrarily among the

(j

divisors of p-1.

§ 133. To obtain a function belonging to the group just considered, we start with the cyclical function belonging to s i, q,2 , .••• qik must then be so taken that the pk different systems of indices z 1 , z2, ... zk give rise to pk different systems q, 1 , q,2, ... qik. These considerations could be further extende.- 2 or less by H 1 • If sr 1 is a function belonging to H 1 , then 9'1 is determined from '? by an Abelian equation of degree pa-"i, the group of which again contains only substitutions of order p, and so on.

§ 186.

Theoren1 XVI.

The solution of an irreducible Abelian equation of degree pa., the group of which contains only substitutions of, order p and the identical substitution, reduces to that of a irreducible Abelian equations of deg1·ee p. Although this Theorem is contained as a special case iu that obtained in § 183, we will again verify it by the aid of the method last employed. Let s 1 be any substitution of the group G of the given Abelian ·equation; then the order of s 1 is p. Again if s2 is any substitution of G not contained among the powers of s 1 , then since s1 s2 = 82 s1 , the group H = j s1 , s2 f contains at the most p 2 substitutiorn,. It will contain exactly this number, if the equality si"s/ = s1a.sl requires that a= a, b = (1. But if si"s} = stsi, then st-a.= s/1-\ and for every value of /i- b different from O we can determine a number m such that mp-b) __ l (mod. p). It follows then that 82 = 82m(fJ-b) = st•(a-a. \ which is contrary to hypothesis. Accordingly /J = b and a= a. If a> 2, suppose that s 3 is a substitution of G not contained .among the p 2 substitution sts/. Since s 1s 3 = 8 38 1 and B.Sa = 83S2, the group H 3 = {s 1 , s2 , s3 f contains at the most p 3 substitutions. And it contains exactly this number, for if sts/sa" = B1 4 slsaY, then .s3-r-c =st-a.s/-fl, and so on, as before. Proceeding in this way, we perceive that all the substitutions of G can be written in the form. s/'•s.J'• ... s/·a., ().,= 0, 1, .. . p-l)

218

THEORY OF SUBSTITUTIONS.

where every substitution occurs once and only once (cf. § 183). now we take for the resolvents and the corresponding groups

If

e: • I

t.hen every resolvent depends on an Abelian equation of degree PThe roots of the given equation of degree pa are rational functions. of S"i, So 2 , • • • \Ca, for the function

\!• = ,:i\ ¥'1

+ 11d2+ ... + 1a\l'a 1

belongs to the group 1 ( cf. § 177).

§ 187.

The pa roots of such an equation may be denoted by

(z,\=0, 1, 2, ... p-1) Suppose that .rs,,~,, ... ~a is the root by which s/1s}• ... Na~a replaces, x,,, z,, ... ,., . Then the substitution

by virtue of the left hand form will replace x,, ,.,,, ... =a by the root hy v.thich s,'ii s}, ... s/ia replaces Xs,, ~., ... ~a. But from the right hand form this root is x ~,+ t, , ~ +t, ... sa +ta. Consequently every 1

substitution s/1s}, ... sata replaces any element

xs,.~.---s..

by

that is the substitutions of the group are defined analytically by the formula (mod.p). The group of an .Llbelian equation of degree pa., the SJtbstitutions of i.chich are all of order p, consists of the arithmetic substitutions of degree p" (mod. p).

§ 188. Finally we effect the transition from the investigations of the present Chapter to the more special qum,tions of the preceding one. 2rr Let n be any arbitrary integer and let the quotient - be denoted n by a. Then, as is well known, the n quantities

219

THE ABELIAN EQUATIO~S.

cosa, cos2a, cos3a, ... cosna satisfy an equation, the coefficient'> of which are rational numbers,

C)

Xn _ _l

4

If now we write

..t·

11J.•"- 2

+

= ens a,

n(n-3) v:> 0)

is rational within the rational domain. Since p is a prime number, it is possible to find an integer ,., such that the congruence mr,=1 or the equation UlfJ.

shall be satisfied.

=op+]

Then the quantity

and consequently x', is rational. From the reducibility of the equation would therefore follow the rationality of a root, which is certainly impossible. The group of the equation is of order p(p-1 ). For if we leave one root x 1 unchanged, any other root 11JX1 can still be converted into any one of the p-1 roots wx1 , ulx1 , w"x, ... wP- 1x 1 •

RATIONAL RELATIONS BETWEEN THREE ROOTS.

Theorein JV.

229

'l'he binomial equation

xP-A=O, in ichich A is nnt the p power of any q11ctntity bdouging to the rational domain, belonas to the type of ~ 196. Its woiip i,q of m·der th

p(p-1). ~ 1\.l8. REMARK. By Theorem III every irreducible equation the roots of which are rational functions of two among them i,; algebraically solvable. At present we.have not the means of proving the converse theorem. It will however be shown in the following Chapter by algebraic consideration,;, and again at a latei· period in the treatment of ,;olvable equation,; by the aid of the theory of groups, that C'I.Jei·y eqnatio11 of pri1ne degree, u.:hich i.~ irredHrif.Jle and algebraically soli-able, is either an equation of the typl' a/1oce c011,,idP.1wl, or an A helian eqzwtion. Before we pa,;s to such general considerations, we treat first another special case, characterized by rational relations among the roots taken three by three.

§ 10ll. An equation is ,;aid to be of h'iad character, or it is called briefly a friad equation,* if its root,; can Le arranged in triads xa, x 13 , .ry in such a way that any two elements of a t.riad determine the third element rationally, i. e., if ,ra and .t·13 determine.{'-},, x 13 and Xy determine Xa, and a·y and Xa determine .1·13. Thus the equations of the third degree are triad equations; for ..r1 +.x·2+Xs = Va-2, • • • V2, Vii go, g2, · .. gP-1"

The new value assumed by x 0 is then fo=go+g1v1+g2v/+gav/+ ... +gP_ 11,t- 1. From § 218 .;0 is again a root, and this together with the system ~p-1> which arises from ~o when v 1 is replaced by wvH w 2v 1 , • • • wP- 1vi, gives again the complex of all the roots. We can therefore take ;'i, ;'2 • • • •

+g2V/ +••.=Go+ w' V1 + G2w' 2v.2+ ... , go+wvl +g2w2v/+ ... =Go+ w"V1+ G2wmv,2+ ... , go+ w2V1 +g2w'v/+ ••.=Go+ w'" Vi+ G2w'mvi2-t- ... , go+v1

where w', (/)",(I}"', ... are the p th roots of unity w, w2, w3, ... , apart. from their order. By addition of these equations we obtain go= Go, so that G0 is unaffected by the modifications of Va-u Va-z, ... V 2_ Also p G0 is the sum of all the roots, and is therefore a rational function in the domain (ffi', fil", ... ). Again we obtain from the system above the equation pv1=Go(l+w- 1+ llJ- 2 )+ Vi(llJ' +w"w- 1+w"'w- 2+ .. . ) + G2V/(1,/2+ w"2 m- 1+w'mw- 2+ .. . ),

+ ...

+ ... Here the first term on the right vanishes. We denote the parentheses in the following terms briefly by pQ u pQ 2 , pf!. 3 , • • • , and write

9) On raising this to the p th power

A)

vt=F1(v2,v8, ... v,._ 1;ffi', ... )=[!.l1V1+!.l2G2V.2+ ... J"' =A0+A 1V 1+A2V/+ ... •

and annexing the equation of definition

B) it. -follows from· Theorem I that either V1 is rat.ional in or that

258

THEORY OF SUBSTITUTIONS.

We consider now the first of these alternatives. In the rational expression of V1 in terms of V 2 , V3 , ••• ; v2 , Va, ••. ; ffi' ffi", ... all the v 2 , Va, ••• Va_ 1 cannot vanish; otherwise V 2 should have been suppressed in 2). If then we define V1 , as in §§ 208 and 212 by a system of successive radicals, some V." will occur last among the v's and some V>.. last among the v's. If we substitute the expression for V1 in x 0 , we have Xo =R(Vi, ... Va-l;!Ji', .. . )= R1(V2, · .. Va-ii Ve, ... Va-1;ffi', .. . )

Here all the v's cannot vanish, as we have just seen. For the same reason all the V's cannot vanish, since we might have started out from ~o- But VK and v,. are two external radicals, and the product of their exponents must therefore be a factor of p (Theorem V). This being impossible, the first alternative is excluded. Accordingly we must have in A) V/ = A 0 ,

A 1= 0,

A 2 = 0, ... AP- I = 0.

The question now arises what the form of 0) must be in order that its p th power may take th~ form Vi"= A 0 • The equation A) is vt=A0 +A1V1+A2V/+ ... =[.Q1V1+.!22G2V/+ .. . y.

The result just obtained shows that the left member is unchanged "if V 1 is replaced by wVi, w2 V/,... Consequently

+ !!2G2w2V/+ ... y = ut, [.Q1w2V1 + .Q2G2w'V/ + ... ]P = v/,

[!!1 w V1

and on the extraction of the p th root we have U1wV1+.Q2G2w 2V/+ ... =:wKvl.

But from 9) follows also QlwKVI +n2G2wKvi2+ ... =wKV1,

and equating the two left members and applying Theorem I as usual, we have

a1=0, U2G2=0, ... .QK_IGK-1=0, aK+1GK+1=0, ... , that is, 0) reduces to the single term

0')

THE ALGEBRAIC SOLUTION OF EQUATIONS.

Substituting this result, together with g0 = G0 in the expression for f;o, we have

.;o = Go+.QKGK Vt+gz(Q.GK Vt)2+ aa(Q.G. Vt)3+ ... On the other hand the root ~o, which is contained among x 0 , x 1 , can also be expressed in the form f0 =

••• ,

G0 + w'V1 + G2 w' 2 V/+ G3 w' 3 V/+ ... ,

and, comparing the two right members, it follows from Theorem I that terms with equal or congruent exponents (mod. p) are identical. In particular we have and therefore

10)

Theoren1 VIII. If, in th~ explicit expression 3) of the root .x0 of an irreducible equution of prime deg1·ee p, the frrationalities V are modified in any way consistent icith the equations of definition, then Vt is coiwerted ctt the isame time into (G;.. V/)P,.

§ 221. 'l'he relations arising from such a transformation are most readily discussed by the introduction of a pnmiffoe congruence mot e in the place of the prime number p. \Ve write V1

= J:/ R

0,

G, Vi'= ,{I Ru

G,, V/ = ,{I R.i, ... ,

where however every ea is to be reduced to its least not negative remainder (mod. p-1). Then the quantities V1,

Ga V13. • • • Gp-I vt- 1

G2 V1 2,

-coincide, apart from their order, with the quantities

,{/ Ro, ,V R 0

,{/

R2, ...

J:/ Rp-2,

and we have 11)

x0 = G0

+ ,{I R + ,V R, + ... +,{I R 0

0 _2•

The changes in the values of the radicals, considered above, which replace Vt by (G;.. V 1")P and consequently [Ga Via] by [Ga;.. Vi""Y, where a;.. < p and a;..=).a (mod.p), have therefore the effect of replacing every Ra by

2130

THEORY OF SUBSTITUTIONS.

Ra+K

where xis

< p-1

(a=0,1, ... p-2),

and is defined by the congruence eK=).

(mod. p).

Consequently the quantities

I) are converted in order into

and, if the same operation is performed

rl

times, I) is replaced by

R.,K, RaK+I• RaK+2• • • • +RaK+p-2,

where the indices are of course to be reduced (mod. p-1). If there is another modification of the radicals, which converts R 0 into Rµ., this on being repeated {J times converts the series I) into R13µ., R13µ.+1, R13,,.+2, · · · R13µ.+p-2•

Finally if we apply the first operation a times and the second fi times, I) becomes

+

Here rt and /I can be so chosen that az /J_,,_ gives the greatest common divisor of x and:,. Consequently if Rk is the R of lowest indt>x which is obtainable from R 0 by alteration of the radicals, every other R obtainable from R 0 in this way will have for its index a ·multiple of k, so that the permutations of the R's take place only within the systems

I

Here k is a divisor of p-1. There are thfln alterations in the meaning of the radicals which produce the substitution

(Ro Rk R2k • • • ) (R1 Rk+I R2k+1 · • • ) · · • § 222. The preceding developm8Ilts enable ns to determine the group of the irreducible solvable equations 1) of prime degree p. Every permutation of the x's can only be produced by the alter-

THE ALGEBRAIC SOLUTION OF EQUATIONS,

261

ations in the radicals V 1 , V 2 , ••• Va_,, and consequently only such permutations of the x's can occur in the group as are produced by alterations of the V's. From the result of the preceding Sectinn

v/,

V 1 can be converted into wTG,1< and the possible alterations in Substituting this in the table of § 215, we have fo = Uo wTG.k v/"

Vi do not change this form.

+

~I=

+ ... ' Go+wT+ 1aek v/ + ... ,

We examine now whether any root Xµ can remain uncµanged m this transformation. In that case we must have

and from the method whir.h we have repeatedly employed it follows, as a necessary and sufficient condition, that

p.ek _/,.

+

T

(mod. p).

If ek= 1 (mod. p), then for T : 0 there is no solution 11., and therefore no root x µ which remains unchanged. But for T = 0, every 11. satisfies the condition, and the substitution reduces to identity. If e1" ~~ 1, then for every T there is a single solution 11., and the corresponding substitution leaves only one element unchanged.

Theore1n IX. The group of a solvable irreducible equation of prime degree is the metacyclical g1·oup (§ 13-i) or one of its subgroups. § 223. Since now, as we saw in § 221, all the substitutions of the group permute the values R 0 ,Rk,R2k, ... only among themselves, the symmetric functions of th~se values are known, and the values them-

p-1

selves are the roots of an equation of degree -,:;--- .

The latter is an

Abelian equatiou since the group permutes thevalues R 0 , Rk,R2k, ... only cyclically. Consequently every Rk, R 2k, Rsk, ... is a rationa function of R 0 • But the same is true of every R.,. For the form of the substitution at the end of S 221 shows that after the adjunc-

262

THEORY OF SUBSTITUTIONS.

t10n of R 0 all the other Ra.'s. are known, since the group reduces to 1. Finally it appears that if Ra.=Fa.(Ro) then

Ra+km = Fo.(Rkm}•

For the application of properly chosen substitutions of the group convert~ the first equation into the second. We consider now all the substitutions of the group of f(x)= 0 which leave Ro= V{ unchanged and accordingly can only convert V 1 into some w• V1 • Then x 0 is replaced by x,. But since R 1 is a rational function of R 0 , it appears that R 1 G/Vep is also unchanged, so that G. Vi' is converted into some G. w,,. V/. The power wP- can be determined from x,; for the expression for x, contains the term G. V/w"", and this must be identical with G,w,,. V1'. Consequently µ = ,e, and G. Vi" becomes

=

while at the same time Vi" becomes

w""Vi", so that the factor G, remains unchanged. That is, every substitution of the group, which leaves R 0 unchanged, leaves G. unchanged also. Accordingly G. is a rational function of Ro- The same is true of all the other G's. vV e can therefore write

12) x 0 = G0 + V,+ SoiV{)· V/+ q,a(Vt)- V/+ ... + q,P_i(V{)· V{- 1, where q,2 , % , . . . are rational functions of Vt in the domain (ffi', ffi", ... ). From this it appears that in 11) the radicals ,(,I R 0 , ,{-' R 1 , . • • do not admit of multiplying every term by an arbitrary root of unity, as indeed is already evident d priori since otherwise x 0 would have not p, but pP values. A still further transformation of 12) is possible. ·we have

l/ R1 = G. 1./ R = /ii(Ro) · ,V Ro•• 0•

1

From § 221 there are alterations in the V 1 , V 2 , ••• Va._ 1 which convert R 0 into Rk and consequently ,{,/ R 0 into ,,,,,,k 1./ Rk. The form of the exponent of w evidently involves no limitation. At the same time the x 0 becomes

263

THE ALGEBRAIC SOLUTION OF EQUATIONS.

x.,= Go+w".,(.f Ro+• .. +

,,,vekJ:.,f R,. +wvek+1,{/ Rk+1+ • •.'

and since R1 becomes R,.+ 1 , it follows that W

VR

1

becomes

c,ek+l'P/-R 'V

k+J•

If now we apply these transformations to the equations above, we obtain

wv•k+l 1.,1 Rk+l = c/•1(R,.). (w"'llo ,V Rk)', ,{/ Rk+1 = c/•1(R,.) • ~ R/. We can therefore also write

+t/R,., +VR2,.,+ ••. +cJ• 1(Ro) · {I Ro"+cfl1(R,.) · ;,/ R,.'+cf• 1(R2 k) • ,{/ R2k•

Xo=Go+ ~Ro

+ .. . + t/•z(Ro) · 1./ R/2 + c/•2(R,.) · ,V R,.-2 + PaYJJ.8 •

•• ,

JJ3> P,8 ••

-,

Vv e make use of the theorem of § 128, and write r = p. aq, where then p 1 > q. G contains at least one subgroup H of the order pt. If we denote by kp, 1 the total number of subgroups of order pt contained in G1 and by p.ai the order of the maximal subgroup of G which is commutative with H, then 1· = p/ i(kPi 1). Since r = p."-q and q < p 1 , we must take k = 0 and r = Pi "-i. That is, G is itself commutative with H. By the solution of an auxiliary equation of degree q, with a group of order q, we arrive therefore at a function belonging to the family of H, and the group G reduces to H(§ 232), Theorem X). From Theorem VII the latter group is solvable. Accordingly, if the auxiliary equation is solvable, the group G is soJvable also. The g1·oup of the auxiliary equation with the order q = Pl p 3Y ••• admits of the same treatment as G. Its solvability therefore follows

+

+

• L. Sylow: Math, ADIi, V, p. 585-

290

THEORY OF SUBSTITUTIONS.

from that of a new auxiliary equation wifh a group of order p 3"1p,6 • •• , and so on. § 243. We return t,o the general investigations of § 241. The transition frogi G to G, decomposes the Galois resolvent . mto . equat10n -r = p, f actors. The transition from G1 to G2 decomr,

poses each of these previously irreducible factors inte

'0 = p r2

2

new

factors, and so on. Since f(x) = 0 was originally irreducible, but is finally resolved into linear factors, it follows from § 235 that once or oftener a resolution of f(x) or of its already rationally known facto1·s will occur simultaneously with the resolution of the Galois resolvent equation or of its already known rational factors. The number of factors into which f(x) = 0 resnlves, which is of course greater than 1, must from § 235, be a divisor of the number of factors into which the Galois resolvent equation divides. In the case of solvable equations the latter is abvays a prime number Pu p 2 , p 3 , . . . Consequently the same is true of f(x) = 0. All prime factors of the degree n of the solvable equation f (x) = 0 are factors of composition of the group G, and in fact each factor occurs in the serie.~ of composition as often as it occurs in n. To avoid a natural error, it must be noted that if in passing from G to G>. the polynomial f(x) resolves into rational factors one of which is f' >.(x), this factor does not necessarily belong to the group G>.. It may belong to a family included in that of G>.The number of values of f' >.(x) is therefore not necessarily equal to r: 1'>.- It may be a multiple of this quotient. And the product f' >.(x) -f" >.(x) . . . of all the values of f\(x) is not necessarily equal to .f(x), but may be a power of this polynomial. We will now assume that n is not a power of a prime number p, so that n includes among its factors different p1·ime numbers. Then different prime numbers also occur among the factors of composition of the series for G, and consequently (§ ij4, Corollary I) G has a p1·incipal series G, H, J, K, ... M, I. Suppose that in one of the series of composition belonging to G other groups

ALGEBRAICALLY SOLVABLE EQUATIONS.

291

H', H", ... H(>..)

3)

occur between H and J. Since n includes among its factors at least two different prime numbers, f(x) must resolve into factors at least twice in the passage from a group of the series of composition to the following one. Since the number of the factors of f(x) is the same as the factor of composition, and since the latter is the same for all the intermediate groups 3), the two reductions of f(x) cannot both take place in the same transition from a group H of the principal series to the next following group J. It is to be particularly noticed, that all the resolutions of f(x) cannot occur in the transition from the last group M to 1, that is, within the groups M', M", ... Af(K- 1), 1,

following M in the series of composition. At least one of the resolutions must have happened before M. Suppose, for example, that the first resolution occurs between H' and H". Then it follows from § 235 that H' is non-primitive in those elements which it connects transitively, and that H" is intransitive, the systems of int~ansitivity coinciding with the system of non-transitivity of H'. The same intransitivity then occurs in all the followiilg groups H'", ... H(>..J, and likewise in the next group J of the principal series, which by assumption is different from 1. Suppose that J distributes the roots in the intransitive systems

x' 1 ,x'2 ,

•••

x';;

x'' 1 ,x'' 2 ••• x'';;

these systems being taken as small as possible.

Then the expression

f' >.(x) = (x-x'i) (x-x\) ... x-x';) becomes a rationally known factor of f(x), which does not contain any smaller rationally known factor. Since from the properties of the groups of the principal series

a-

1

JG=J,

all the values off' >.(x) belong to the same group J. They are therefore all rationally known with f'>.(x). Of the values of .f'>..(x) we know already

f\(x) = (x-x'i) (x-x 12 ) ••• (x-x',), f''>. (x) = (x-x''i) (x-x'\) ... (x-x",),

292

THEORY OF SUBSTITUTIONS.

If there were other values, these must have roots in common with some J;,,(a.l (x) and consequently f' A(x) would resolve into rational factors. This being contrary to assumption, .f' A(x) has

only m

m.

= 1~i

values, and is therefore a root of an equation of degree

If this equation is

'f(Y)-(y-J'A) (y-J"A) • .. (y-Ji.,_(m)) = 0,

4)

then f(x) is the result of elimination between 4) and

5)

f'A(x)=x;-¢•,(y')x1-1+¢•ly')x;-2 __ .. =0,

where ¢• 1(y')

= x' 1+ x\ +x' + ... + x',. 3

+ ... +c2!;.(Al+a2-0,

b1:;/>->

+ ... + c ,:_(A)+ a1==0, 1

O (A= ,1, ... x)

have only one solution each, viz: a 1 = 1, b1 = 0, ... c1 = 0; a 1 = 0, ~ = 0, b2 = 1, ... C 2 = 0; a 2 = 0,

and these solutions furnish together tthe identical substitution 1. We designate now a system of "+ 1 roots of an equation fo'r which E_0 (mod. p) as a system of conjugate roots. We have then

Theorem XV. If a substitution of a primitive solvable group of degree p• leaves unchanged "+ 1 roots which do not form a conjugate system, the substitution reduces to identity.

301

ALOEBllAICALLY SOLVABLE EQUATIONS.

+

If therefore we adjoin x 1 such roots to the equation, the group G reduces to those substitutions which leave z 1 roots unchanged, i. e., to the identical substitution. The equation is then solved.

+

Theore1n XVI. All the roots of a solvable primitive equation of degree p• can be rationally expressed in terms of any x 1 among them, provided these do not f