Proof of a Conjecture of Wiegold

TL;DR

This paper proves a conjecture by Wiegold, establishing a relationship between the size of the derived subgroup of a finite p-group and the minimum breadth of its generators.

Contribution

It confirms Wiegold's conjecture by showing that large derived subgroup size implies the existence of generators with high breadth in finite p-groups.

Findings

If |G'| > p^{n(n-1)/2}, then G can be generated by elements with breadth at least n.

The proof links the size of the derived subgroup to the breadth of generators.

Provides a new criterion for generating finite p-groups based on subgroup size.

Abstract

In this short note we confirm a conjecture of James Wiegold. We prove that if $G$ is a finite $p$-group and $|G'|>p^{n(n-1)/2}$ for some non-negative integer $n$, then the group $G$ can be generated by the elements of breadth at least $n$. The breadth $b(x)$ of an element $x$ of a finite $p$-group $G$ is defined by the equation $|G:C_G(x)| = p^{b(x)}$, where $C_G(x)$ is the centralizer of $x$ in $G$.