Powers in finite groups and a criterion for solubility

Martin W. Liebeck, Aner Shalev

Research output: Contribution to journalArticlepeer-review


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.

Original languageAmerican English
Pages (from-to)4179-4189
Number of pages11
JournalProceedings of the American Mathematical Society
Issue number12
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • General Mathematics


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