On products of conjugacy classes in general linear groups

Preusser, Raimund

doi:10.22108/ijgt.2021.123469.1627

تعداد نشریات	43
تعداد شماره‌ها	1,651
تعداد مقالات	13,405
تعداد مشاهده مقاله	30,225,198
تعداد دریافت فایل اصل مقاله	12,079,974

	On products of conjugacy classes in general linear groups
International Journal of Group Theory
مقاله 8، دوره 11، شماره 4، اسفند 2022، صفحه 229-252 اصل مقاله (489.87 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2021.123469.1627
نویسنده
Raimund Preusser^*
Chebyshev Laboratory, St. Petersburg State University, Russia
چکیده
Let $K$ 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 $n=\infty$.
کلیدواژه‌ها
general linear groups؛ conjugacy classes؛ matrix identities

مراجع
[1] Z. Arad, J. Stavi and M. Herzog, Powers and products of conjugacy classes in groups, Products of conjugacy classes in groups, Lecture Notes in Math., 1112, Springer, 1985 6–51. [2] H. Bass, K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. ´ , 22 (1964) 5–60. [3] A. Beltran, M.J. Felipe, C. Melchor, Some problems about products of conjugacy classes in finite groups, Int. J. Group Theory 9 (2020) 59–68. [4] P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic abstract algebra, Basic abstract algebra, Second edition, Cambridge University Press, Cambridge, 1994. [5] J. L. Brenner, The linear homogeneous group. III, Ann. of Math. (2), 71 (1960) 210–223. [6] E. W. Ellers, N. Gordeev and M. Herzog, Covering numbers for Chevalley groups, Israel J. Math., 111 (1999) 339–372. [7] I. Z. Golubchik, The full linear group over an associative ring, Uspehi Mat. Nauk, 28 (1973) 179–180. [8] N. Gordeev, Products of conjugacy classes in algebraic groups I, II, J. Algebra, 173 (1995) 715–744, 745–779. [9] R. Guralnick, G. Malle and P. Huu Tiep, Products of conjugacy classes in finite and algebraic simple groups, Adv.Math., 234 (2013) 618–652. [10] F. Kn¨uppel and K. Nielsen, The extended covering number of SLn is n+ 1, Linear Algebra Appl., 418 (2006) 634–656. [11] A. Lev, Products of cyclic conjugacy classes in the groups PSL, Linear Algebra Appl., 179 (1993) 59–83. [12] A. Lev, Products of cyclic similarity classes in the group GLn(F), Linear Algebra Appl., 202 (1994) 235–266. [13] A. Lev, The covering number of the group PSLn(F), J. Algebra, 182 (1996) 60–84. [14] R. Preusser, Sandwich classification for GLn(R), O2n(R) and U2n(R, Λ) revisited, J. Group Theory, 21 (2018) 21–44. [15] R. Preusser, Reverse decomposition of unipotents over noncommutative rings I: General linear groups, Linear Algebra Appl. 601 (2020) 285–300. [16] D.M. Rodgers, J. Saxl, Products of conjugacy classes in the special linear groups, Comm. Algebra, 31 (2003) 4623–4638. [17] L. N. Vaserstein, On the normal subgroups of GLn over a ring, Lecture Notes in Math., 854 (1981) 454–465. [18] L. N. Vaserstein, Normal subgroups of the general linear groups over Banach algebras, J. Pure Appl. Algebra, 41 (1986) 99–112. [19] L. N. Vaserstein, Normal subgroups of the general linear groups over von Neumann regular rings, Proc. Am. Math. Soc, 96 no. 2 (1986) 209–214. [20] L. N. Vaserstein and E. Wheland, Products of conjugacy classes of two by two matrices, Linear Algebra Appl., 230 (1995) 165–188. [21] J. S. Wilson, The normal and subnormal structure of general linear groups, Proc. Cambridge Philos. Soc., 71 (1972) 163–177.
آمار تعداد مشاهده مقاله: 786 تعداد دریافت فایل اصل مقاله: 219

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

On products of conjugacy classes in general linear groups