Some properties of schemes in group theory and Top couples

TL;DR

This paper generalizes a topology on prime ideals in group theory, aiming to develop new tools for solving classical problems by exploring properties like functor of points, dimension, and Galois theory in this context.

Contribution

It extends the topology on prime ideals from G-groups to broader classes of groups, introducing new concepts and potential methods for group theory research.

Findings

Defined a topology on prime ideals for various group classes

Introduced notions of functor of points and dimension in this topology

Outlined a Galois theory framework for these topological structures

Abstract

Let G be a group, and H a G-group defined by an imbedding map $G\rightarrow H$; in [12] we have defined a topology on a subset of normal subgroups of $H$, the so-called prime ideals. In this work, we generalize this topology to other classes of groups. We hope that such topologies define new effective tools to tackle well-known problems in groups theory. We study properties of the topology defined in [12], for example we define the notions of functor of points, dimension and outline a Galois theory.