On the class-breadth conjecture

TL;DR

This paper investigates the class-breadth conjecture for p-groups with odd primes, exploring whether the nilpotency class is bounded by the breadth plus one, extending prior work mainly focused on p=2.

Contribution

The paper advances understanding of the class-breadth conjecture for p > 2, providing new insights or results in this open area of group theory.

Findings

Counter-examples for p=2 are known.

The conjecture remains open for p > 2.

This work explores the conjecture in the case p > 2.

Abstract

The class-breadth conjecture of Leedham-Green, Neumann and Wiegold states that for each $p$-group, $cl(G)\leq b(G) + 1$, where $cl(G)$, $b(G)$ denote the nilpotency class and the breadth of $G$. While several counter-examples to this conjecture have been found for $p = 2$, it is still open in general for $p > 2$. This article is dedicated to the general case $p > 2$ of the conjecture.