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

Anil Kumar, Chudamani Pranesachar; Pradhan, Soham Swadhin

doi:10.22108/ijgt.2021.129815.1708

تعداد نشریات	43
تعداد شماره‌ها	1,675
تعداد مقالات	13,674
تعداد مشاهده مقاله	31,689,941
تعداد دریافت فایل اصل مقاله	12,519,612

	On the endomorphism semigroups of extra-special $p$-groups and automorphism orbits
International Journal of Group Theory
مقاله 6، دوره 11، شماره 4، اسفند 2022، صفحه 201-220 اصل مقاله (489.17 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2021.129815.1708
نویسندگان
Chudamani Pranesachar Anil Kumar¹؛ Soham Swadhin Pradhan^* ²
¹School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi 211019, Prayagraj, INDIA
²Department of Mathematics, Postdoctoral fellow, Harish-Chandra Research Institute, India
چکیده
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 $End(G)$.
کلیدواژه‌ها
Extra-special $p$-Groups؛ Heisenberg Groups؛ Automorphism Groups؛ Endomorphism Semigroups؛ Symplectic Groups

مراجع
[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.
آمار تعداد مشاهده مقاله: 847 تعداد دریافت فایل اصل مقاله: 391

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

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