Multiplicative groups avoiding a fixed 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).