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 $p^2$. This article mainly describes the endomorphism semigroups of b...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2022-12-01
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| Series: | International Journal of Group Theory |
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| Online Access: | https://ijgt.ui.ac.ir/article_25994_b2f6b3b410dc74377be13112ffda6657.pdf |
| _version_ | 1828129581287079936 |
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| author | Chudamani Pranesachar Anil Kumar Soham Swadhin Pradhan |
| author_facet | Chudamani Pranesachar Anil Kumar Soham Swadhin Pradhan |
| author_sort | Chudamani Pranesachar Anil Kumar |
| collection | DOAJ |
| description | 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 $p^2$. 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 $End(G)$. |
| first_indexed | 2024-04-11T16:20:48Z |
| format | Article |
| id | doaj.art-452784b141224f38bc7940eb95113c97 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-04-11T16:20:48Z |
| publishDate | 2022-12-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-452784b141224f38bc7940eb95113c972022-12-22T04:14:21ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692022-12-0111420122010.22108/ijgt.2021.129815.170825994On the endomorphism semigroups of extra-special $p$-groups and automorphism orbitsChudamani Pranesachar Anil Kumar0Soham Swadhin Pradhan1School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi 211019, Prayagraj, INDIADepartment of Mathematics, Postdoctoral fellow, Harish-Chandra Research Institute, IndiaFor 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.pdfextra-special $p$-groupsheisenberg groupsautomorphism groupsendomorphism semigroupssymplectic groups |
| spellingShingle | Chudamani Pranesachar Anil Kumar Soham Swadhin Pradhan On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits International Journal of Group Theory extra-special $p$-groups heisenberg groups automorphism groups endomorphism semigroups symplectic groups |
| title | On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits |
| title_full | On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits |
| title_fullStr | On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits |
| title_full_unstemmed | On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits |
| title_short | On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits |
| title_sort | on the endomorphism semigroups of extra special p groups and automorphism orbits |
| topic | extra-special $p$-groups heisenberg groups automorphism groups endomorphism semigroups symplectic groups |
| url | https://ijgt.ui.ac.ir/article_25994_b2f6b3b410dc74377be13112ffda6657.pdf |
| work_keys_str_mv | AT chudamanipranesacharanilkumar ontheendomorphismsemigroupsofextraspecialpgroupsandautomorphismorbits AT sohamswadhinpradhan ontheendomorphismsemigroupsofextraspecialpgroupsandautomorphismorbits |