Published online by Cambridge University Press: 21 December 2018
Let $I(n)$ denote the number of isomorphism classes of subgroups of
$(\mathbb{Z}/n\mathbb{Z})^{\times }$, and let
$G(n)$ denote the number of subgroups of
$(\mathbb{Z}/n\mathbb{Z})^{\times }$ counted as sets (not up to isomorphism). We prove that both
$\log G(n)$ and
$\log I(n)$ satisfy Erdős–Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that
$\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of
$\log G(n)$ and
$\log I(n)$.