# Intersection of maximal subgroups in prosolvable groups

## TL;DR

This paper explores the structure of prosolvable groups by examining how intersections of maximal subgroups, called η-intersections, influence the group's properties and what they reveal about the group's composition.

## Contribution

It introduces the concept of η-intersections in prosolvable groups and analyzes their implications for the group's structural characteristics.

## Key findings

- Characterization of η-intersections in prosolvable groups
- Relationship between η-intersections and group structure
- Insights into subgroup intersection properties

## Abstract

Let $H$ be an open subgroup of a profinite group that can be expressed as intersection of maximal subgroups of $G.$ Given a positive real number $\eta,$ we say that $H$ is an $\eta$-intersection if there exists a family of maximal subgroups $M_1,\ldots,M_t$ such that $H=M_1\cap\ldots\cap M_t$ and $|G:M_1|\cdots|G:M_t|\le |G:H|^{\eta}$. We investigate the meaning of this property and its influence on the group structure.