Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Commuting probabilities of finite groups   The commuting probability of a finite group is defined to be the probability
that two randomly chosen group elements commute. Let P \subset (0,1] be the set
of commuting probabilities of all finite groups. We prove that every point of P
is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce
two conjectures of Keith Joseph from 1977: all limit points of P are rational,
and P is well ordered by >. We also prove analogous theorems for bilinear maps
of abelian groups.

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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: hep-lat