A topological interpretation about $m_{G, N}$ for finite group $G$ with normal subgroup $N$

TL;DR

This paper explores a topological perspective on the invariant $m_{G,N}$ of finite groups with a normal subgroup, linking it to the Euler characteristic of nerve spaces derived from a class poset.

Contribution

It introduces a new topological interpretation of $m_{G,N}$ using class posets and nerve spaces, establishing a relation with their Euler characteristics.

Findings

Established a relation between $m_{G,N}$ and Euler characteristics of nerve spaces.

Constructed a class poset of $G$ based on cyclic subgroups.

Provided a topological framework for analyzing group invariants.

Abstract

Let $G$ be a finite group and $N\unlhd G$. In this note, we construct a class poset of $G$ for some cyclic subgroup $C$ of $G$. And we find a relation between $m_{G,N}$ and the Euler characteristic of some nerve spaces of these posets(see Main Theorem).