Multiplicative groups avoiding a fixed group

TL;DR

This paper provides an asymptotic formula for counting integers whose multiplicative groups do not contain a given finite abelian group as a subgroup, highlighting that such groups appear in almost all multiplicative groups.

Contribution

It establishes an asymptotic count for integers with multiplicative groups avoiding a fixed finite abelian subgroup, extending understanding of subgroup appearances in multiplicative groups.

Findings

Almost all multiplicative groups contain any fixed finite abelian group as a subgroup.

An explicit asymptotic formula for the count of integers with multiplicative groups avoiding a given subgroup.

The result applies to all finite abelian groups except the trivial one-element group.

Abstract

We know that any finite abelian group $G$ appears as a subgroup of infinitely many multiplicative groups $\mathbb{Z}_n^\times$ (the abelian groups of size $\phi(n)$ that are the multiplicative groups of units in the rings $\mathbb{Z}/n\mathbb{Z}$). It seems to be less well appeciated that $G$ appears as a subgroup of almost all multiplicative groups $\mathbb{Z}_n^\times$. We exhibit an asymptotic formula for the counting function of those integers whose multiplicative group fails to contain a copy of $G$, for all finite abelian groups $G$ (other than the trivial one-element group).