Counting the Closed Subgroups of Profinite Groups

TL;DR

This paper investigates the structure and classification of the spaces of closed and closed-normal subgroups in profinite groups, combining algebraic and topological techniques to determine their size and classify them up to homeomorphism.

Contribution

It provides a detailed calculation of the size of subgroup spaces and offers a partial classification up to homeomorphism, advancing understanding of their topological and algebraic properties.

Findings

Calculated the size of the spaces of closed and closed-normal subgroups.

Partially classified these subgroup spaces up to homeomorphism.

Combined algebraic and topological methods for analysis.

Abstract

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.