On the maximal cardinality of half-factorial sets in cyclic groups

We consider the function μ(G), introduced by W. Narkiewicz, which associates to an abelian group G the maximal cardinality of a half-factorial subset of it. In this article, we start a systematic study of this function in the case where G is a finite cyclic group and prove several results on its behaviour. In particular, we show that the order of magnitude of this function on cyclic groups is the same as the one of the number of divisors of its cardinality.