On varietal capability of Infinite direct products of groups

Recently , the authors gave some conditions under which a direct product of finitely many groups is mathcal{V}-capable if and only if each of its factors is mathcal{V}-capable for some varieties mathcal{V} . In this paper , we extend this fact to any infinite direct product of groups . Moreover , we conclude some results for mathcal{V}-capability of direct products of infinitely many groups in varieties of abelian , nilpotent and polynilpotent groups .

Keywords:

Capable group , Direct product , Variety of groups , V−capable group , direct limit