  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.
