On varietal capability of Infinite direct products of groups

Abstract

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.