Corollary of Timmesfeld's replacement theorem for abelian subgroups

This article is about a maximality notion among subgroups, related to abelianness or small class, in a group of prime power order.
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Statement

Suppose $P$ is a group of prime power order. Let ${\mathcal {A}}(P)$ denote the set of abelian subgroups of maximum order in $P$. If $A\in {\mathcal {A}}(P)$, and $V$ is an $A$-invariant abelian subgroup of $P$. Then, $[V,A]C_{A}([V,A])$ is an abelian subgroup of maximum order.

Related facts

• Timmesfeld's replacement theorem
• Corollary of Timmesfeld's replacement theorem for elementary abelian subgroups

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 copyMore info}