Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the proportion of $p$-elements in a finite group, and a modular
  Jordan type theorem   In 1878, Jordan proved that if a finite group $G$ has a faithful
representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal
abelian subgroup with index bounded above by a function of $n$. The same result
fails if one replaces $\mathbb{C}$ by a field of positive characteristic, due
to the presence of large unipotent and/or Lie type subgroups. For this reason,
a long-standing problem in group and representation theory has been to find the
"correct analogue" of Jordan's theorem in characteristic $p>0$. Progress has
been made in a number of different directions, most notably by Brauer and Feit
in 1966; by Collins in 2008; and by Larsen and Pink in 2011. With a 1968
theorem of Steinberg in mind (which shows that a significant proportion of
elements in a simple group of Lie type are unipotent), we prove in this paper
that if a finite group $G$ has a faithful representation over a field of
characteristic $p$, then a significant proportion of the elements of $G$ must
have $p$-power order. We prove similar results for permutation groups, and
present a general method for counting $p$-elements in finite groups. All of our
results are best possible.

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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
0
The categories of this paper are: hep-ph