## Half-factorial sets in elementary $$p$$-groups.(English)Zbl 1120.20055

A subset $$G_0$$ of an Abelian group $$G$$ is called half-factorial if all factorizations of an element in the semigroup $$B(G_0)$$, consisting of zero-sum sequences of elements of $$G_0$$ with juxtaposition as multiplication, are of the same length. The author studies half-factorial subsets in elementary $$p$$-groups and determines the structure of such sets with largest cardinality $$\mu(G)$$. In the case when either $$p\leq 7$$ or the rank $$r$$ of $$G$$ is $$\leq 2$$, the equality $$\mu(G)=1+[r/2](p-2)+r$$ is established. The author conjectures that this formula holds for all elementary $$p$$-groups.

### MSC:

 20K01 Finite abelian groups 11R27 Units and factorization 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20M14 Commutative semigroups