On the Number of Not Powers in a Finite Group

Let G be a finite group and let k be a positive integer. We examine the relationship between structural properties of G and the number of elements of G that are not kth powers in G. In particular, we examine a bound on |G| given by Lucido and Pournaki and classify all cases when it is strict. We also show that when k is an odd prime, then either G has a normal subgroup with specific properties, or |G| is bounded above by a tighter function dependent on the number of not k-th powers of G.