Counting multiplicative groups with prescribed subgroups

TL;DR

This paper investigates the asymptotic counts of integers with specific multiplicative group structures, linking group theory problems to prime factor restrictions and applying classical analytic number theory techniques.

Contribution

It provides explicit asymptotic formulas for counting integers with prescribed Sylow subgroup structures and maximally non-cyclic groups, connecting group theory with prime factor restrictions.

Findings

Asymptotic count for integers with a given Sylow q-subgroup structure.

Asymptotic count for integers with maximally non-cyclic multiplicative groups.

Reduction of group-theoretic problems to prime factor restriction counting.

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 $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $K x(\log\log x)^\ell/(\log x)^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$. Second, we consider the set of integers $n\le x$ such that the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ 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 $A x/(\log x)^{1-\xi}$ for an explicit constant $A$, where $\xi$ 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.