The pre-periods of all endomorphisms of a finite cyclic group

Let G be a group. By Fitting’s lemma, if G is finite abelian and t is the pre-period of G, then G can be represented as the direct sum of kerf t and f t (G) for all endomorphisms f of G. In this paper, we will show that the pre-period of a cyclic group of order n is the maximum of νp (n) (the p-adic valuation of n) for all prime p with p|n. Indeed, the pre-periods of all endomorphisms of a finite cyclic group are determined in terms of the p-adic valuations.