A relation between $m_{G, N}$ and the Euler characteristic of the nerve space of some class poset of $G$

TL;DR

This paper establishes a connection between a group-theoretic invariant $m_{G,N}$ and the Euler characteristic of a nerve space derived from a class poset of a finite group, enabling direct computation of $m_{G,N}$.

Contribution

It introduces a new method to compute $m_{G,N}$ using the Euler characteristic of a nerve space associated with a class poset, providing a direct computational approach.

Findings

Computed $m_{S_5, A_5}=0$ directly.

Established a relation between $m_{G,N}$ and the Euler characteristic.

Proved $S_5$ is a $B$-group.

Abstract

Let $G$ be a finite group and $N\unlhd G$ with $|G: N|=p$ for some prime $p$. In this note, to compute $m_{G,N}$ directly, we construct a class poset $\mathfrak{T}_{C}(G)$ of $G$ for some cyclic subgroup $C$. And we find a relation between $m_{G,N}$ and the Euler characteristic of the nerve space $|N(\mathfrak{T}_{C}(G))|$ (see the Theorem 1.3). As an application, we compute $m_{S_5, A_5}=0$ directly, and get $S_5$ is a $B$-group.