Decomposition of groups and top couples

Abstract

We have endowed various categories of groups with topologies [12]. The purpose of this paper is to introduce on these categories others topologies which are more suitable to study well-known problems in groups theory. We use this framework to define a notion of prime ideal and to provide a decomposition of a large class of groups into a product of prime ideals. Remark that a similar question has been studied in [5] by Kurata. We remark that these topologies can be extended to other categories like the categories of commutative algebras, associative algebras and left symmetric algebras.