Finitely generated Abelian $n$-groups

In this paper, the isomorphism of a finitely generated Abelian $n$-ary group and of a direct product of a finite number of indecomposable Abelian semi-cyclic $n$-ary groups, being partly finite primary and partly infinite ones, is proved. A complete system of invariants for finitely generated Abelian $n$-ary groups is found. We point out a necessary and sufficient condition for a direct product of infinite Abelian semi-cyclic $n$-ary groups to be a free $n$-ary group in the class of Abelian $n$-ary groups.