On the class-breadth conjecture for $p$-groups

Abstract

The class-breadth conjecture of Leedham-Green, Neumann and Wiegold states that the nilpotency class of any $p$-group is at most $b(G)+1$, where $b(G)=\underset{g\in G}{\max }{\log }_p[G:Z_G(g)]$ denotes the breadth of a group $G$. 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.