# Central intersections of element centralisers

## TL;DR

This paper explores a new structural classification of groups based on intersections of element centralisers, extending the concept of CA-groups and applying similar ideas to F-groups, leading to broader classifications.

## Contribution

It introduces a weakened intersection condition for centralisers that generalizes CA-groups and applies similar methods to classify F-groups, expanding the understanding of group structures.

## Key findings

- New classification of groups based on centraliser intersections
- Generalization of CA-group structure
- Extension of classification methods to F-groups

## Abstract

In 1970 R. Schmidt gave a structural classification for CA-groups. In this paper we consider a condition upon the intersection of element centralisers which turns out to be equivalent to the definition of a CA-group. We then weaken which centralisers we chose to intersect and structurally classify this new family of groups. Furthermore we apply a similar weakening to the class of F-groups introduced by Ito in 1953 and classified by Rebmann in 1971.