Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Explicit constructions of loops with commuting inner mappings   In 2004, Cs\"{o}rg\H{o} constructed a loop of nilpotency class three with
abelian group of inner mappings. Until now, no other examples were known. We
construct many such loops from groups of nilpotency class two by replacing the
product $xy$ with $xyh$ in certain positions, where $h$ is a central
involution. The location of the replacements is ultimately governed by a
symmetric trilinear alternating form.

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The categories of this paper are: quant-ph
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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: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas