  In recent work, Pomerance and Shparlinski have obtained results on the number
of cycles in the functional graph of the map $x \mapsto x^a$ in
$\mathbb{F}_p^*$. We prove similar results for other families of finite groups.
In particular, we obtain estimates for the number of cycles for cyclic groups,
symmetric groups, dihedral groups and $SL_2(\mathbb{F}_q)$. We also show that
the cyclic group of order $n$ minimizes the number of cycles among all
nilpotent groups of order $n$ for a fixed exponent. Finally, we pose several
problems.
