In recent work, Pomerance and Shparlinski have obtained results on the number of cycles in the functional graph of the map x !→ xa in 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 SL2(Fq). 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 a. Finally, we pose several problems.
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