  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.

Number Theory