Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Structure of Commutative Automorphic Loops   An \emph{automorphic loop} (or \emph{A-loop}) is a loop whose inner mappings
are automorphisms. Every element of a commutative A-loop generates a group, and
$(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and
$p$ a prime. The loop $Q$ has order a power of $p$ if and only if every element
of $Q$ has order a power of $p$. The loop $Q$ decomposes as a direct product of
a loop of odd order and a loop of order a power of 2. If $Q$ is of odd order,
it is solvable. If $A$ is a subloop of $Q$ then $|A|$ divides $|Q|$. If $p$
divides $|Q|$ then $Q$ contains an element of order $p$. If there is a finite
simple nonassociative commutative A-loop, it is of exponent 2.

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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin