Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Covers of Dihedral 2-Groups by Powerful Subgroups   A finite $p$-group $G$ is called \textit{powerful} if either $p$ is odd and
$[G,G]\subseteq G^p$ or $p=2$ and $[G,G]\subseteq G^4$. A {\em{cover}} for a
group is a collection of subgroups whose union is equal to the entire group. We
will discuss covers of $p$-groups by powerful $p$-subgroups. The size of the
smallest cover of a $p$-group by powerful $p$-subgroups is called the
\textit{powerful covering number}. In this paper we determine the powerful
covering number of the dihedral 2-groups.

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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin