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… 

Tables from this paper

Shintani descent, simple groups and spread

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

Maximal Cocliques in the Generating Graphs of the Alternating and Symmetric Groups

The generating graph $\Gamma(G)$ of a finite group $G$ has vertex set the non-identity elements of $G$, with two elements connected exactly when they generate $G$. A coclique in a graph is an empty

The Non-Commuting, Non-Generating Graph of a Nilpotent Group

TLDR
It is shown that if $G$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi+G$ in the diameter $3$ case and to any group with every maximal subgroup normal.

Forbidden subgraphs in generating graphs of finite groups

Let G be a 2-generated group. The generating graph Γ(G) is the graph whose vertices are the elements of G and where two vertices g1 and g2 are adjacent if G = 〈g1, g2〉. This graph encodes the

Graphs defined on groups

‎This paper concerns aspects of various graphs whose vertex set is a group $G$‎ ‎and whose edges reflect group structure in some way (so that‎, ‎in particular‎, ‎they are invariant under the action

A note on the invariably generating graph of a finite group.

We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components of order at least $2$.

CONNECTED COMPONENTS IN THE INVARIABLY GENERATING GRAPH OF A FINITE GROUP

  • Daniele Garzoni
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 2021
Abstract We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

Graphs encoding the generating properties of a finite group

Assume that G is a finite group. For every a,b∈N , we define a graph Γa,b(G) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,⋯,xa) and (y1,⋯,yb) are adjacent if and

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

...