Group Theory by Hartmut Laue

By Hartmut Laue

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Example text

Then P ∩ G′ = P ∩ Q′ |Q ∈ Sylp (G) [P, NG (P )]. Proof. Put P ∗ := P ∩ Q′ |Q ∈ Sylp (G) . We show, by induction on o(g), that every g ∈ P ∩ G′ is contained in P ∗ [P, NG (P )]. This is trivial for g = 1G . For the inductive step let g ∈ P ∩ G′ and o(g) > 1. 3, we have −1 P ∗ = gvG→P/P ∗ = P ∗ (g |B| )xB , −1 (g |B| )xB ∈ P ∩ G′ for all P -orbits B in G /. P. B −1 −1 If |B| > 1, then o((g |B| )xB ) < o(g), hence inductively (g |B| )xB ∈ P ∗ [P, NG (P )]. As |B| = 1 if and only if xB ∈ NG (P ), it follows that −1 g[g, x−1 B ] = g |NG (P ):P | ≡ B∈NG (P )/P −1 g xB ≡ (g |B| )xB ≡ 1G mod P ∗[P, NG (P )], B |B|=1 hence g ∈ P ∗ [P, NG (P )] because p ∤ |NG (P ) : P |.

X = n where n ∈ N. 36 The elements of U ≀ M have the form (u, (m1 , . . , mn )) (where u ∈ U, mj ∈ M) which is the product f (u, (1M , . . , 1M )) (1U , (m1 , . . , mn )). No confusion will arise if we write this product simply as u(m1 , . . , and their canonical embeddings in U ≀ M. We f know that conjugation of an element of M n by an element of u ∈ U is given by −, i. , executing ← u ∀m1 , . . , mn ∈ M ∀u ∈ U ← − (m1 , . . , mn )u = (m1 , . . , mn ) u = (m1u−1 , . . , mnu−1 ). Thus the i-th component of an n-tuple over M is sent to the (iu)-th position.

0                For each j, the element mj is pulled down to the i-th place in the jth column, where i = jπ −1 . All other matrix entries are 0. Thus the index of the row containing the element mj is determined by π. Conversely these indices clearly determine π, while the n-tuple (m1 , . . , mn ) is trivially read from the matrix (by “ignoring all zeros”). In particular, Φ is injective. Every matrix in the image of Φ has exactly one entry different from 0 in every row and in 36 Recall that, by definition, an n-tuple over M is a function of n into M .

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