Timmesfeld's replacement theorem

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This article defines a replacement theorem
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Suppose G is a finite group, p is a prime number, and V is a faithful \mathbb{F}_p(G)-module, i.e., a vector space over the field of p equipped with a G-action. Consider the set s_p(G,V) of elementary abelian p-subgroups A of G such that |A||C_V(A)| \ge |B||C_V(B)| for all subgroups B of A. Suppose A \in s_p(G,V). Then, we have:

  • C = C_A([V,A]) is nontrivial.
  • |C||C_V(C)| = |A||C_V(A)|. In particular, C \in s_p(G,V).
  • |C_V(C)| = [V,A]C_V(A).

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