# Timmesfeld's replacement theorem

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## Statement

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, $C \in s_p(G,V)$.
• $|C_V(C)| = [V,A]C_V(A)$.