Powers in finite groups and a criterion for solubility

Liebeck, Martin; Shalev, Aner

doi:10.1090/S0002-9939-2013-11790-9

Powers in finite groups and a criterion for solubility
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by Martin W. Liebeck and Aner Shalev PDF

Proc. Amer. Math. Soc. 141 (2013), 4179-4189 Request permission

Abstract:

We study the set $G^{[k]}$ of $k^{th}$ powers in finite groups $G$. We prove that if $G^{[12]}$ is a subgroup, then $G$ must be soluble; moreover, 12 is the minimal number with this property. The proof relies on results of independent interest, classifying almost simple groups $G$ and positive integers $k$ for which $G^{[k]}$ contains the socle of $G$.

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