The sequence of non k-th powers of a finite group G

Abstract

We examine the connection between the sequence of not k-th powers of a finite group G and structural information about G. It is known that a single nonzero entry in the sequence bounds the order of G. We show that information about the number of specific not k-th powers can determine whether G is nilpotent, has a normal Hall subgroup, and other structural information about G. Furthermore, we show that the set of not k-th powers uniquely determines finite abelian groups up to isomorphism.

Keywords:

• Finite group
• finite Abelian group
• nilpotent
• not k-th powers

2010 Mathematics Subject Classification:

• 20E20
• 20K20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1747503. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.