On the class-breadth conjecture

TL;DR

Let $cl(G)$ denote the nilpotency class and $b(G)$ the breadth of a finite $p$-group $G$; the paper investigates the conjecture $cl(G)\le b(G)+1$ for odd primes $p$. It develops a formal framework with detailed notations for commutators, centralizers, and lower central series, and proves a collection of lemmas that provide fine-grained control over normal subgroups and their interactions in $p$-groups. A strengthened conjecture is introduced, positing an $(b(G)+2)\times(b(G)+2)$ upper triangular ladder of subgroups $(A_{ij})$ from which $\gamma_{b(G)+2}(G)=\{e\}$ follows, and the original conjecture is shown to follow from this statement. The paper confirms the strengthened conjecture for a special class of $p$-groups (where each normal non-abelian $H$ yields a noncyclic $H/[H,G]$ and a prescribed commutator containment), using an inductive construction that handles abelian and non-abelian layers and outlines a path toward a general proof.

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.

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof