Intersection graphs of quasinormal subgroups of general skew linear groups

Abstract

The intersection graph of quasinormal subgroups of a group $G$, denoted by $\Gamma_{\mathrm{q}}(G)$, is a graph defined as follows: the vertex set consists of all nontrivial, proper quasinormal subgroups of $G$, and two distinct vertices $H$ and $K$ are adjacent if $H\cap K$ is nontrivial. In this paper, we show that when $G$ is an arbitrary nonsimple group, the diameter of $\Gamma_{\mathrm{q}}(G)$ is in $\{0,1,2,\infty\}$. Besides, all general skew linear groups $\mathrm{GL}_n(D)$ over a division ring $D$ can be classified depending on the diameter of $\Gamma_{\mathrm{q}}(\mathrm{GL}_n(D))$.

Keywords

• division ring
• general skew linear group
• intersection graph
• quasinormal subgroup
• permutable subgroup

Supporting Institution

Vietnam National University HoChiMinh City (VNUHCM)

Project Number

T2022-18-03

Thanks

This research is funded by Vietnam National University HoChiMinh City (VNUHCM) under grant number T2022-18-03.

References

1. [1] S. Akbari, F. Heydari and M. Maghasedi, The Intersection Graph of a Group, J. Algebra its Appl. 14(05), Article No. 1550065 (9 pages), 2015.
2. [2] S. Akbari, R. Nikandish, M. J. Nikmehr, Some Results on the Intersection Graphs of Ideals of Rings, J. Algebra its Appl. 12(4), Article No. 1250200 (13 pages), 2013.
3. [3] S. Akbari, H. Tavallae, S. K. Ghezelahmad, Some Results on the Intersection Graph of Submodules of a Module, Math. Slovaca 67(2), 297-304, 2017.
4. [4] M. H. Bien and D. H. Viet, Intersection Graphs of General Linear Groups, J. Algebra its Appl. 20(03), Article No. 2150039 (12 pages), 2021.
5. [5] J. Bosak, The Graphs of Semigroups (in Theory of Graphs and Application), Academic Press-New York, 119125, 1964.
6. [6] A. Cayley, Desiderata and Suggestions: No. 2. The Theory of Groups: Graphical Representation, Am. J. Math. 1, 174-176, 1878.
7. [7] S. K. Chebolu and K.Lockridge, Fuchs Problem for Indecomposable Abelian Groups, J. Algebra 438, 325336, 2015.
8. [8] L. Q. Danh, M. H. Bien and B. X. Hai, Permutable Subgroups in $\mathrm{GL}_n(D)$ and Applications to Locally Finite Group Algebras, Vietnam J. Math. 53(2), 277–288, 2023.

9. [9] L. Q. Danh and H. V. Khanh, Locally Solvable Subnormal and Quasinormal Subgroups in Division Rings, Hiroshima Math. J. 51, 267-274, 2021.
10. [10] P. K. Draxl, Skew Fields (London Mathematical Society Lecture Note Series 81), Cambridge University Press, 1983.
11. [11] C. Faith, Algebraic Division Ring Extensions, Proc. Amer. Math. Soc. 11(1), 43–43, 1960.
12. [12] F. Gross, Infinite Permutable Subgroups, Rocky Mt. J. Math. 12(2), 333-343, 1982.
13. [13] I. N. Herstein, Multiplicative Commutators in Division Rings, Isr. J. Math. 31(2), 180-188, 1978.
14. [14] M. Mahdavi-Hezavehi, Commutators in Division Rings Revisited, Bull. Iran. Math. Soc. 26(2), 7-88, 2000.
15. [15] V. Ramanathan, On Projective Intersection Graph of Ideals of Commutative Rings, J. Algebra its Appl. (20)(2), Article No 2150017, (16 pages), 2021.
16. [16] D. J. S. Robinson, A Course in the Theory of Groups (Graduate Texts in Mathematics), Springer, 1995.
17. [17] W.R.Scott, Group Theory, Dover Publications, Inc. New York, 1987.
18. [18] S. E. Stonehewer, Permutable Subgroups of Infinite Groups, Math. Z. 125, 1-16, 1972.
19. [19] B. A. F. Wehrfritz, Soluble Normal Subgroups of Skew Linear Groups, J. Pure Appl. Algebra 42(5), 95107, 1986.
20. [20] B. A. F. Wehrfritz, Soluble and Locally Soluble Skew Linear Groups, Arch. Math. 49(5), 379388, 1987.