Proof of a Conjecture of Wiegold

In this short note we confirm a conjecture of James Wiegold. We prove that if $G$ is a finite $p$-group and $\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }>p^{n(n-1)\mathrm{/}2}$ 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 $\mathrm{\mid }G:C_G(x)\mathrm{\mid }=p^{b(x)}$, where $C_G(x)$ is the centralizer of $x$ in $G$.