Bouc's conjecture on $B$-groups

TL;DR

This paper investigates Bouc's conjecture relating the nilpotency of a finite group G to its largest quotient B-group, proving the conjecture for certain non-solvable groups with specific simple group factors.

Contribution

It extends Bouc's conjecture verification to non-solvable groups with a single nonabelian simple group factor, excluding certain Chevalley groups.

Findings

Proves Bouc's conjecture for groups with one simple factor not among specified Chevalley groups.

Shows that if G has a unique simple factor S (not in the excluded list), then β(G) is not solvable or nilpotent.

Establishes the conjecture holds in these specific non-solvable cases.

Abstract

Bouc proposed the following conjecture: a finite group $G$ is nilpotent if and only if its largest quotient $B$-group $\beta(G)$ is nilpotent. And he has prove that this conjecture holds when $G$ is solvable. In this paper, we consider the case when $G$ is not solvable. Let $S$ be a nonabelian simple group except the Chevalley groups $A_{n}(q)$, $D_{n}(q)$, $E_{6}(q)$, and $^2A_{n}(q)$, if there exists only one factor of $G$ which is isomorphic to $S$, then $\beta(G)$ is not solvable, of course, is not nilpotent. That means we prove the conjecture in these cases.