Timmesfeld's replacement theorem

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

Related facts

 * Thompson's replacement theorem for abelian subgroups
 * Glauberman's replacement theorem
 * Corollary of Timmesfeld's replacement theorem for elementary abelian subgroups