Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coprime invariable generation and minimal-exponent groups   A finite group $G$ is \emph{coprimely-invariably generated} if there exists a
set of generators $\{g_1, ..., g_u\}$ of $G$ with the property that the orders
$|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \in G$
the set $\{g_1^{x_1}, ..., g_u^{x_u}\}$ generates $G$.
  We show that if $G$ is coprimely-invariably generated, then $G$ can be
generated with three elements, or two if $G$ is soluble, and that $G$ has zero
presentation rank. As a corollary, we show that if $G$ is any finite group such
that no proper subgroup has the same exponent as $G$, then $G$ has zero
presentation rank. Furthermore, we show that every finite simple group is
coprimely-invariably generated.
  Along the way, we show that for each finite simple group $S$, and for each
partition $\pi_1, ..., \pi_u$ of the primes dividing $|S|$, the product of the
number $k_{\pi_i}(S)$ of conjugacy classes of $\pi_i$-elements satisfies
$\prod_{i=1}^u k_{\pi_i}(S) \leq \frac{|S|}{2| Out S|}.$

9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas