Some necessary conditions for compatibility of groups

TL;DR

This paper establishes new necessary conditions for the compatibility of two groups, which involves the existence of a finite group with isomorphic normal subgroups leading to the given groups as quotients.

Contribution

It introduces novel necessary conditions that must be satisfied for two groups to be compatible, advancing the understanding of group compatibility criteria.

Findings

New necessary conditions for group compatibility

Characterization of compatibility via normal subgroups

Implications for the structure of finite groups

Abstract

Two groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. In this paper, we give new necessary conditions for two groups to be compatible.