Decomposition of groups and Top couples

Recently, we have endowed various categories of groups with topologies. The purpose of this paper is to introduce on these categories others topologies which are statistically 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 Remark that a similar question has been studied in [5] by Kurata with innocent methods. We remark that these topologies can be extended to other categories like the categories of commutative algebras, associative algebras and left symmetric algebras.