Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bol loops and Bruck loops of order $pq$ up to isotopism   Let $p>q$ be odd primes. We classify Bol loops and Bruck loops of order $pq$
up to isotopism. When $q$ does not divide $p^2-1$, the only Bol loop (and hence
the only Bruck loop) of order $pq$ is the cyclic group of order $pq$. When $q$
divides $p^2-1$, there are precisely $\lfloor(p-1+4q)(2q)^{-1}\rfloor$ Bol
loops of order $pq$ up to isotopism, including a unique nonassociative Bruck
loop of order $pq$.

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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an