# On Varietal Capability of Infinite Direct Products of Groups

## TL;DR

This paper extends the understanding of when infinite direct products of groups are a0a0a0a0 capable within certain varieties, generalizing previous finite cases and exploring specific varieties.

## Contribution

It generalizes conditions for a0a0a0a0 capability from finite to infinite direct products of groups across various varieties.

## Key findings

- Extended capability conditions to infinite products.
- Results for abelian, nilpotent, polynilpotent varieties.
- Clarified the structure of a0a0a0a0 capability in infinite contexts.

## 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.