# New PDF release: Sets of Conjugate Cycles of a Substitution Group

By Miller G. A.

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**Example text**

NX(D)nF(X));D. As Thus t inverts Note that a Sylow 2-subgroup of F(X) lies in CX(t) n F(X) ~ D. Now U = [t, U] centralizes NX(D) n F(X);D. Arguing for each Sylow p-subgroup of the nilpotent group NX(D) n F(X) separately, it follows that U centralizes NX(D) n F(X). [U, F(X)] Thus D = F(X) and = 1. Now any Sylow 2-subgroup of U centralizes F(O(X)) ~ F(X). 2 it follows that any Sylow 2-subgroup of U centralizes O(X). 1 it follows that any Sylow 2-sub- group of U lies in F*(X) and so U;; F*(X). 3(a), U = [t, U] <]<] E(X).

First if 0p(X) "* 1, let X = X/Op(X). By induction [t, U]<]

Proof. H' is a p'-group. Suppose that a E P and that a g E P for some g E G. 9, we have -1 Z, zg - 1 ECG (a). There exists x E CG(a) such that (z, zg x) is a p- -1 group. Thus we can find y E G such that (z, zg x)Y;;; P. 11. so g-l x EH. g X Then a = a -1 g and x- 1 g EH. H ' a p'-group. completes the proof of Lemma 7. 12. 1/ transfer. Lemma 7. 13. H' I. This Hence H is a uniqueness subgroup for A. Proof. For let S = pr:O normalizes E or IBI = In I (H). By Lemma 7. 8, NH(S) either p,p (Z2(F(H)q))1 =q3.