On the T$^\circ$-slices of a finite group

TL;DR

This paper introduces the concept of T$^ ightarrow$-slices in finite groups, extending properties of finite groups to these slices and establishing the existence of a largest quotient T$^ ightarrow$-slice for any slice.

Contribution

It defines T$^ ightarrow$-slices in finite groups and proves that every slice has a largest quotient T$^ ightarrow$-slice, extending group properties to this new framework.

Findings

Introduction of T$^ ightarrow$-slices in finite groups

Existence of a largest quotient T$^ ightarrow$-slice for any slice

Extension of properties from finite groups to slices

Abstract

A slice (G, S) of finite groups is a pair consisting of a finite group G and a subgroup S of G. In this paper, we show that some properties of finite groups extend to slices of finite groups. In particular, by analogy with B-groups, we introduce the notion of T$^\circ$-slice, and show that any slice of finite groups admits a largest quotient T$^\circ$-slice.