Counting multiplicative groups with prescribed subgroups

Abstract

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime [Formula: see text] and a finite abelian [Formula: see text]-group [Formula: see text], we consider the set of integers [Formula: see text] such that the Sylow [Formula: see text]-subgroup of the multiplicative group [Formula: see text] is isomorphic to [Formula: see text]. We show that the counting function of this set of integers is asymptotic to [Formula: see text] for explicit constants [Formula: see text] and [Formula: see text] depending on [Formula: see text] and [Formula: see text]. Second, we consider the set of integers [Formula: see text] such that the multiplicative group [Formula: see text] is “maximally non-cyclic”, that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to [Formula: see text] for an explicit constant [Formula: see text], where [Formula: see text] is Artin’s constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.