Power maps in finite groups

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 ${\mathbb{F}}_p^{\ast }$. 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 $S{L}_2({\mathbb{F}}_q)$. 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.