Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $G = \langle x_i, y\rangle$ for all $i$. The
more restrictive notion of uniform spread, denoted $u(G)$, requires $y$ to be
chosen from a fixed conjugacy class of $G$, and a theorem of Breuer, Guralnick
and Kantor states that $u(G) \geqslant 2$ for every non-abelian finite simple
group $G$. For any group with $u(G) \geqslant 1$, we define the uniform
domination number $\gamma_u(G)$ of $G$ to be the minimal size of a subset $S$
of conjugate elements such that for each nontrivial $x \in G$ there exists $y
\in S$ with $G = \langle x, y \rangle$ (in this situation, we say that $S$ is a
uniform dominating set for $G$). We introduced the latter notion in a recent
paper, where we used probabilistic methods to determine close to best possible
bounds on $\gamma_u(G)$ for all simple groups $G$.
  In this paper we establish several new results on the spread, uniform spread
and uniform domination number of finite groups and finite simple groups. For
example, we make substantial progress towards a classification of the simple
groups $G$ with $\gamma_u(G)=2$, and we study the associated probability that
two randomly chosen conjugate elements form a uniform dominating set for $G$.
We also establish new results concerning the $2$-generation of soluble and
symmetric groups, and we present several open problems.

1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci