On the Invariance of the Factors of Composition of a Substitution-GroupAuthor(s): James PierpontSource: American Journal of Mathematics, Vol. 18, No. 2 (Apr., 1896), pp. 153-155Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369679 .

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On the Invariance of the Factors of Composition of a Substitution-group.

BY JAMES PIERPONT, New Havene, G7onn.

The theorem* which asserts the invariance of the factors of composition of a group is of great importance in the theory of algebraic equations. It may be stated thus:

In whatever manner we may decompose a group into a.series of com- position

G, Gil G2, G3, .. G,= 1,

the factors of composition a,, a2, as, . . X a,

are in every case the same, aside from their order. 9 Jordan's lengthy demonstration of this theorem has been greatly simplified

by Nettot in his "1 Treatise on the Theory of Substitutions "; but by employing the notion of isomorphism the proof may be made still more simple.4

The conception of isomorphism may be arrived at as follows: Let the equa- tion F(x) 0 have for a certain region of rationality the group G. Let cpl be any rational function of the roots of F (x) = 0 belonging to a subgroup G, of G, which for G takes on the values

Any substitution of G operating on this series will, in general, cause the P's to permute among themselves, and thus give rise to a group of substitutions r

* C. Jordan, Trait6 des Substitutions et des Ic-quations Algebriques. Paris, 1870, p. 41-48. f E. Netto, Substitutionentheorie und ihre Anwendung auf die Algebra. Leipzig, 1882, p. 87-90. t Cf. J. K6nig, Ueber rationale Functionen, etc. Mathematische Annalen, vol. XIV, p. 212-30. 0. Holder, Zuriickfnhrung einer beliebigen algebraischen Gleichung auf eine Kette von Gleich-

ungen. Math. Ann., vol. XXXIV, p. 26-56.

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154 PIERPONT: On the Invariance of the Factors

which, having the same properties as those of G, are said to be isomorphic to G. To a substitution of G will correspond a substitution of r, while to every substi- tution of r will correspond one and the same nuniber of substitutions of G, namely, the number of substitutions of G corresponding to the identical substi- tution 1 of r.

The proof of (A) depends upon the following lemma:

Let the group G of order r have two maximal invariant subgroups H (order h, index a) and K (order k, index B). The group of substitutions C (order c) common to H and X is a maximal invariant subgroup for H S (B) and K. The index a of C under H is equal to 3, while its index r under | K is equal to a. J

We proceed then to prove (B). In the first place we notice that C is invariant for H and K. For S= H-1 CH

is a substitution of H or K according as we regard C as in H or K. Thus S is common to Hand K, and hence is in C, so that H-1 CH= C.

Let now cp be a rational function of the roots of F(x) 0 O belonging to C, and let r = (71, 72 .0.) be the group of substitution in the p's corresponding to G; while A = (Q, 32....) and 8 = (1, 32 . . ) may correspond to H and K resp. Then A and 0- are maximal invariant subgroups of r of orders a and 't

resp. The order of r is p- -

C

The groups A and 8) have only the identical substitution in common; for if a be common to them, then the corresponding substitutions 9,, S2 a ** of G are common to H and K and are hence in C. But to the substitutions of C corres- pond the identical in r.

The substitutions A, S are commutative. For the substitution A 13- S is equal to the identical substitution, since it is common to A and 8), whence

S S= 3. (a)

The substitutions of r are all of the form

7-SS. (b)

For these substitutions form a subgroup of r wllich is invariant for r, since A and 8 are. As this group contains the maximal subgroups A and' 8, it is iden- tical with r. Further, from S = 5 follows S-I1 = 35, which, being a sub- stitution common to A and 8, is the identical substitution; whence - =j and , = 5i.Thusp = ar.

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of Composition of a Substitution-group. 155

Let A - (x1, ?2 . . .) be an invariant subgroup of A; then by (a) it is also invariant for e3. The substitution7ta = Xa a2 form a subgroup II - (7z,, n2 * - - -) of r by virtue of (a), which is invariant for r, since A and A are. To II will correspond the invariant subgroup P of G. As II contains ?3, P will contain Kf, which being a maximal invariant subgroup of G requires that A 1, so that C is a maximal invariant subgroup for A or O.

Finally since r = cr _=ca= = cq3t; a-=, I =. The demonstration of (A) now follows easily as by Netto.

Let G, G1, G2, G3 (1)

and G, H1, H2, 13... (2)

be two series of composition for G, having the factors of composition

al,, a2 , 0C3 . o o o (a) and 1%' f2 3 *l' ** (1?)

Then if r be the substitution of G common to G, and H1, the series

G, G1,rF.... (3) G, H r,P.... (4)

having by virtue of (B) the sanme factors, the demonstration of the identity of (a), (3) is reduced to proving that the factors of (1) are the same as those of (3); also that the factors of (2) and. (4) are alike. But while (1), (2) have only the group G in common, (1) and (3) have e. g. two groups, G and G1 in coinmon. Applying the same reasoning to (1), (3), we obtain two new series having three groups in common, and so on. Finally, we need to show that if one mode of decomposition of a certain group L be

L,M, 1

while another is L, R .....

then these two series have the same factors. But this is evident from (B).

21

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