University of IsfahanInternational Journal of Group Theory2251-765011420221201On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits2012202599410.22108/ijgt.2021.129815.1708ENChudamani PranesacharAnil KumarSchool of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi 211019, Prayagraj, INDIA0000-0002-3074-8891Soham SwadhinPradhanDepartment of Mathematics, Postdoctoral fellow, Harish-Chandra Research Institute, India0000-0002-6401-0895Journal Article20210731For 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 $p^2$. This article mainly describes the endomorphism semigroups of both the types of extra-special $p$-groups and computes their cardinalities as polynomials in $p$ for each $n$. Firstly a new way of representing the extra-special $p$-group of exponent $p^2$ is given. Using the representations, explicit formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types are found. 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 $End(G)$.https://ijgt.ui.ac.ir/article_25994_b2f6b3b410dc74377be13112ffda6657.pdf