On the class-breadth conjecture for $p>2$ -groups

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 ${\displaystyle b(G)=\underset{g\in G}{\max }{\log }_p[G:Z_G(g)]}$ denotes 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. We propose a generalization for the case $p>2$, which we prove under some additional conditions.