Burnside Groups by Mennicke J. L.

By Mennicke J. L.

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2, K e r $ = T . 1. 4 Brauer's permutation lemma The aim of this section is to present a variant of a combinatorial result known as Brauer's permutation lemma (Brauer (1941, Lemma 1)). An extension of this result to nonsplitting fields will be presented in the next section. In what follows, F denotes an arbitrary field and G a finite group. As usual, all FG-modules are assumed to be finitely generated. e. I n v ( V ) = {v E V l g v = v for all g E G} We say that V is a permutation module if there exists a basis of V on which G acts as a permutation group.

Let m be divisible by n and let g E G. rn Then gFm = 1 and, since pm E l(modk), we have g;, = gpt. Thus proving (1). Let z = C g E G z g gE FG. 1, = xpm c z:mgPm(mod[FG,FG]) sEG so x E T ( F G ) if and only if CgEGzimgptE [FG,FG]. 3, we infer that 2 E T ( F G ) if and only if c xim = o for all i E { I , . . ,r } gESi Bearing in mind that the required assertion follows. H The crucial case of the following result, namely the case where p is a prime dividing the order of G is due t o Brauer (1935).

X m } is equal t o the number of A orbits of { K l , . ,Jim}. Furthermore, if c h a r F = 0 , then f o r each a € A, the number of elements of { X I , . . ,x m } fixed by a is equal to the number of elements of (IC1,. . ,K m } fixed by a. Proof. We denote by V and W the permutation FA-modules corresponding to the action of A on { X I , . . , x m } and (IC1,. . ,K m } ,respectively. Let pv and pw be the matrix representations of A afforded by V and W with respect to the bases { x i , . .

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