Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Endomorphism Semigroups of Extra-special $p$-groups and
  Automorphism Orbits   For an odd prime $p$ and a positive integer $n$, it is well known that there
are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the
Heisenberg group which has exponent $p$ and the second one is of exponent
$p^2$. In this article, a new way of representing the extra-special $p$-group
of exponent $p^2$ is given. These representations facilitate an explicit way of
finding formulae for any endomorphism and any automorphism of an extra-special
$p$-group $G$ for both the types. Based on these formulae, the endomorphism
semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The
endomorphism semigroup image of any element in $G$ is found and the orbits
under the action of the automorphism group $Aut(G)$ are determined. As a
consequence it is deduced that, under the notion of degeneration of elements in
$G$, the endomorphism semigroup $End(G)$ induces a partial order on the
automorphism orbits when $G$ is the Heisenberg group and does not induce when
$G$ is the extra-special $p$-group of exponent $p^2$. Finally we prove that the
cardinality of isotropic subspaces of any fixed dimension in a non-degenerate
symplectic space is a polynomial in $p$ with non-negative integer coefficients.
Using this fact we compute the cardinality of $End(G)$.

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The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
29
The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc