Abstract
Let k be a positive integer, and suppose that the number of elements of a group G that are not k th powers in G is nonzero but finite. If G is finite, we obtain an upper bound on |G|, and we present some conditions sufficient to guarantee that G actually is finite.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Bannai et al. On the number of elements which are not nth powers in finite groups, Comm. Algebra 17 (1989) 2865–2870
Lévai L., Pyber L.: “Profinite groups with many commuting pairs or involutions”. Arch. Math. 75, 1–7 (2000)
Lucido M. S., Pournaki M. R.: Probability that an element of a finite group has a square root. Colloq. Math. 112, 147–155 (2008)
Liebeck M. W., Shalev A.: Powers in finite groups and a criterion for solubility. Proc. Amer. Math. Soc. 141, 4179–4189 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first author was supported under NSF Grant DGE-1256259.
Rights and permissions
About this article
Cite this article
Cocke, W., Isaacs, I.M. & Skabelund, D. On the number of elements that are not kth powers in a group. Arch. Math. 105, 529–538 (2015). https://doi.org/10.1007/s00013-015-0835-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-015-0835-5