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 $|A||C_{V}(A)|\geq |B||C_{V}(B)|$ 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

References

Journal references

A remark on Thompson's replacement theorem and a consequence by Franz Georg Timmesfeld, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 38,Number 6, Page 491 - 495(Year 1982): This paper proves a result now known as Timmesfeld's replacement theorem. There are two corollaries, the corollary for abelian subgroups and the corollary for elementary abelian subgroups.^{Official copy}^{More info}