Louis Auslander's An Account of the Theory of Crystallographic Groups PDF

By Louis Auslander

Complaints of the yankee Mathematical Society
Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236
Published via: American Mathematical Society
DOI: 10.2307/2035904
Stable URL: http://www.jstor.org/stable/2035904
Page count number: 7

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T . /g 2F forms a subgroup, and hence this extension is split. 1 Lemma (Frattini). Let H be a normal subgroup of a finite group G and let P be a Sylow p-subgroup of H . P /. Proof. Let g be an arbitrary element of G. Then gP g 1 is contained in H and is a Sylow p-subgroup of H . By Sylow’s theorem, gP g 1 D hP h 1 for some h 2 H . P /. 2 Lemma. jH j; jF j/ D 1. Then every extension of H by F splits. 38 Chapter 1. Introduction to finite group theory Proof. f; T / be a factor set of this extension.

1 h. /: (11) Proof. Assume that this extension is split. Then there exists a system of representatives ft 0 . /g 2F which is a subgroup. Since the product t 0 . / t 0 . / lies in the coset and also in this subgroup, it is equal to t 0 . /. 1/ D 1. We define now a function h with the help of equations t 0 . / D h. /t . /; 2 F . Then h. / t. / D h. / t . / h. / t . / D h. / T . h. // f . ; / t . 11/. 1/ D 1. Conversely, if there exists a function h W F ! 11/ holds, then the system of representatives fh.

H by the rule X fQ. / D f . ; /: 2F Summing the equations9 f . ; / D T . f . ; // C f . ; over all / f. ; / jF j f . ; / D T . fQ. // C fQ. / fQ. mod jH j/ gives f . ; / D T . nfQ. // C nfQ. / Now it is clear that we can set h D nfQ. /: nfQ. 3 Theorem (Schur). jH j; jF j/ D 1. Then every extension of H by F splits. Proof. Set n D jF j, m D jH j. Let G be an arbitrary extension of H by F . It is enough to prove that G contains a subgroup of order n. We prove this by induction on m. For m D 1 this is trivial, so let m > 1.

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An Account of the Theory of Crystallographic Groups by Louis Auslander


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