Corollary of Timmesfeld's replacement theorem for abelian subgroups

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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.

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