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}
}
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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References

SHOWING 1-10 OF 83 REFERENCES
Finite groups, 2-generation and the uniform domination number
Let $G$ be a finite $2$-generated non-cyclic group. The spread of $G$ is the largest integer $k$ such that for any nontrivial elements $x_1, \ldots, x_k$, there exists $y \in G$ such that $G =
The Spread of Almost Simple Classical Groups
Every finite simple group can be generated by two elements, and in 2000, Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group every nontrivial element
On the involution fixity of exceptional groups of Lie type
TLDR
Borders on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields are determined by combining results with the Lang-Weil estimates from algebraic geometry.
On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type
We study embeddings of groups of Lie type $H$ in characteristic $p$ into exceptional algebraic groups $\mathbf G$ of the same characteristic. We exclude the case where $H$ is of type
Probabilistic generation of finite simple groups, II
On the uniform spread of almost simple linear groups
Abstract Let G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial
Schur covers and Carlitz’s conjecture
AbstractWe use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). An exceptional polynomialf over a finite field
On a class of doubly transitive groups
THE class u(u> 3) of a doubly transitive group of degree n is, according to Bochert,f greater than \n — § Vw. If we confine our attention however to those doubly transitive groups in which one of the
Descent equalities and the inductive McKay condition for types B and E
...
...