On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits

Document Type : Research Paper

Authors

1 School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi 211019, Prayagraj, INDIA

2 Department of Mathematics, Postdoctoral fellow, Harish-Chandra Research Institute, India

Abstract

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 $End(G)$.

Keywords

Main Subjects

References

[1] D. J. Benson and J. F. Carlson, The cohomology of extraspecial groups, Bull. London Math. Soc., 24 (1992) 209–235.
[2] D. J. Benson and J. F. Carlson, Corrigendum: ‘the Cohomology of Extraspecial groups’, Bull. London Math. Soc., 25 (1993) 498–498.
[3] E. A. Bertram, Some applications of graph theory to finite groups, Discrete Math., 44 (1983) 31–43.
[4] A. Y. M. Chin, On non-commuting sets in an extraspecial p-group, J. Group Theory, 8 (2005) 189–194.
[5] L. Dornhoff, Group representation theory, Part A: Ordinary representation theory, Pure and Applied Mathematics, 7, Marcel Dekker, Inc., New York, 1971 pp. 1–254.
[6] K. Dutta and A. Prasad, Degenerations and orbits in finite abelian groups, J. Combin. Theory Ser. A, 118 (2011) 1685–1694.
[7] D. E. Gorenstein, Finite Groups, AMS Chelsea Publishing, 301 (1968).
[8] R. L. Griess Jr., Automorphisms of Extraspecial Groups and Nonvanishing of Degree 2 Cohomology, Pacific J. Math., 48 (1973) 403–422.
[9] H. Liu and Y. Wang, The automorphism group of a generalized extraspecial p-group, Sci. China Math., 53 (2010) 315–334.
[10] H. Liu and Y. Wang, On non-commuting sets in a generalized extraspecial p-group, Acta Math. Sinica, 55 (2012) 975–980.
[11] H. Liu and Y. Wang, On Non-commuting Sets in Certain Finite p-Groups, Algebra Colloq., 22 (2015) 555–560.
[12] H. Opolka, Projective Representations of Extra-Special p-Groups, Glasgow Math. J., 19 (1978) 149–152.
[13] D. J. S. Robinson, A course in the theory of groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York-Berlin, 1982.
[14] D. L. Winter, The automorphism group of an extraspecial p-group, Rocky Mountain J. Math., 2 159–168.
International Journal of Group Theory
Volume 11, Issue 4 - Serial Number 4
December 2022
Pages 201-220

Files

History

  • Receive Date: 31 July 2021
  • Revise Date: 28 September 2021
  • Accept Date: 29 September 2021
  • Published Online: 01 December 2022

Share

How to cite

Statistics