Powers in finite groups and a criterion for solubility

Martin W. Liebeck; Aner Shalev

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

Powers in finite groups and a criterion for solubility

Martin W. Liebeck
, Aner Shalev

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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.

Original language	English
Pages (from-to)	4179-4189
Number of pages	11
Journal	Proceedings of the American Mathematical Society
Volume	141
Issue number	12
DOIs	https://doi.org/10.1090/S0002-9939-2013-11790-9
State	Published - 2013

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AB - We study the set G[k] of kth 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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Powers in finite groups and a criterion for solubility

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