Determining Frobenius complements from sequences of kth powers

Abstract

For a group G and positive integer k , we define G k = { x k | x ∈ G } . We investigate what structural properties of a finite group G are determined by ( | G k | ) k ∈ N . It has been previously shown that one can determine whether or not a group G has a normal Hall π -subgroup for a set of primes π from ( | G k | ) k ∈ N , thus whether or not G is nilpotent is determined from ( | G k | ) k ∈ N . It is an open question whether or not the solvability of a group G is determined by ( | G k | ) k ∈ N . In this work, we show that whether or not G is a Frobenius group is determined from ( | G k | ) k ∈ N . In the case that G is a Frobenius group, we show that the isomorphism class of the Frobenius complement of G is determined by this sequence of k th powers of G .