# The spread of a finite group

@article{Burness2020TheSO,
title={The spread of a finite group},
author={Timothy C. Burness and Robert M. Guralnick and Scott Harper},
journal={arXiv: Group Theory},
year={2020}
}
• Published 2 June 2020
• Mathematics
• arXiv: Group Theory
A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper…

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