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.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011420221201On co-maximal subgroup graph of $Z_n$2212282599510.22108/ijgt.2021.129788.1732ENManideepaSahaDepartment of Mathematics, Presidency University, KolkataSucharitaBiswasDepartment of Mathematics, Presidency University, KolkataAngsumanDasDepartment of Mathematics, Presidency University, KolkataJournal Article20210806The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK = G$. In this paper, we study and characterize various properties like diameter, domination number, perfectness, hamiltonicity, etc. of $\Gamma(\mathbb{Z}_n)$https://ijgt.ui.ac.ir/article_25995_9a3065467c1401001ec336c7b69f0694.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011420221201On products of conjugacy classes in general linear groups2292522603610.22108/ijgt.2021.123469.1627ENRaimundPreusserChebyshev Laboratory, St. Petersburg State University, Russia0000-0002-4454-9455Journal Article20200611Let $K$ be a field and $n\geq 3$. Let $E_n(K)\leq H\leq GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,\ldots,i_m\in\{\pm 1\}$ such that the product $C^{i_1}\cdots C^{i_m}$ contains all nontrivial elementary transvections. In this article we obtain a sharp upper bound for $m(C)$. Moreover, we determine $m(C)$ for any noncentral $H$-class $C$ under the assumption that $K$ is algebraically closed or $n=3$ or $n=\infty$.https://ijgt.ui.ac.ir/article_26036_36aaaeb1e50575a14d9de67d71f69ab6.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011420221201On the probability of zero divisor elements in group rings2532572605410.22108/ijgt.2021.126694.1664ENHaval M.Mohammed SalihDepartment of Mathematics, Faculty of Science, Soran University , Kawa St, Soran, Erbil, Iraq0000-0002-9853-9682Journal Article20201228Let $R$ be a non trivial finite commutative ring with identity and $G$ be a non trivial group. We denote by $P(RG)$ the probability that the product of two randomly chosen elements of a finite group ring $RG$ is zero. We show that $P(RG)<\frac{1}{4}$ if and only if $RG\ncong \mathbb{Z}_2C_2,\mathbb{Z}_3C_2, \mathbb{Z}_2C_3$. Furthermore, we give the upper bound and lower bound for $P(RG)$. In particular, we present the general formula for $P(RG)$, where $R$ is a finite field of characteristic $p$ and $|G|\leq 4$.https://ijgt.ui.ac.ir/article_26054_2690f3deefb66baa0c7be9888228a25a.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011420221201Classification of the pentavalent symmetric graphs of order $8pq$2592702611010.22108/ijgt.2021.120551.1589ENMasoumehAkbarizadehDepartment of Mathematics, Iran University of Science and TechnologyMehdiAlaeiyanDepartment of Mathematics- Iran University of Science and TechnologyRaffaeleScapellatoDepartment of Mathematics, Politecnico di Milano, Milano, ItalyJournal Article20191215A graph $X$ is symmetric if its automorphism group is transitive on the arc set of the graph. Let $p$ and $q$ be two prime integers. In this paper, a complete classification is determined of connected pentavalent symmetric graphs of order $8pq$.https://ijgt.ui.ac.ir/article_26110_e50b50362e1575a8724e40dc3b44738f.pdf