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.
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