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 $\Gamma_{\mathrm{q}}(\mathrm{GL}_n(D))$.
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
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Springer, 1995.
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Algebra 42(5), 95107, 1986.
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49(5), 379388, 1987.
Year 2024,
Volume: 53 Issue: 2, 392 - 404, 23.04.2024
[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] 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] 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] 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] J. Bosak, The Graphs of Semigroups (in Theory of Graphs and Application), Academic
Press-New York, 119125, 1964.
[6] A. Cayley, Desiderata and Suggestions: No. 2. The Theory of Groups: Graphical
Representation, Am. J. Math. 1, 174-176, 1878.
[7] S. K. Chebolu and K.Lockridge, Fuchs Problem for Indecomposable Abelian Groups,
J. Algebra 438, 325336, 2015.
[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] L. Q. Danh and H. V. Khanh, Locally Solvable Subnormal and Quasinormal Subgroups
in Division Rings, Hiroshima Math. J. 51, 267-274, 2021.
[10] P. K. Draxl, Skew Fields (London Mathematical Society Lecture Note Series 81),
Cambridge University Press, 1983.
[11] C. Faith, Algebraic Division Ring Extensions, Proc. Amer. Math. Soc. 11(1), 43–43,
1960.
[12] F. Gross, Infinite Permutable Subgroups, Rocky Mt. J. Math. 12(2), 333-343, 1982.
[13] I. N. Herstein, Multiplicative Commutators in Division Rings, Isr. J. Math. 31(2),
180-188, 1978.
[14] M. Mahdavi-Hezavehi, Commutators in Division Rings Revisited, Bull. Iran. Math.
Soc. 26(2), 7-88, 2000.
[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] D. J. S. Robinson, A Course in the Theory of Groups (Graduate Texts in Mathematics),
Springer, 1995.
[17] W.R.Scott, Group Theory, Dover Publications, Inc. New York, 1987.
[18] S. E. Stonehewer, Permutable Subgroups of Infinite Groups, Math. Z. 125, 1-16, 1972.
[19] B. A. F. Wehrfritz, Soluble Normal Subgroups of Skew Linear Groups, J. Pure Appl.
Algebra 42(5), 95107, 1986.
[20] B. A. F. Wehrfritz, Soluble and Locally Soluble Skew Linear Groups, Arch. Math.
49(5), 379388, 1987.
Qui Danh, L. (2024). Intersection graphs of quasinormal subgroups of general skew linear groups. Hacettepe Journal of Mathematics and Statistics, 53(2), 392-404. https://doi.org/10.15672/hujms.1249433
AMA
Qui Danh L. Intersection graphs of quasinormal subgroups of general skew linear groups. Hacettepe Journal of Mathematics and Statistics. April 2024;53(2):392-404. doi:10.15672/hujms.1249433
Chicago
Qui Danh, Le. “Intersection Graphs of Quasinormal Subgroups of General Skew Linear Groups”. Hacettepe Journal of Mathematics and Statistics 53, no. 2 (April 2024): 392-404. https://doi.org/10.15672/hujms.1249433.
EndNote
Qui Danh L (April 1, 2024) Intersection graphs of quasinormal subgroups of general skew linear groups. Hacettepe Journal of Mathematics and Statistics 53 2 392–404.
IEEE
L. Qui Danh, “Intersection graphs of quasinormal subgroups of general skew linear groups”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, pp. 392–404, 2024, doi: 10.15672/hujms.1249433.
ISNAD
Qui Danh, Le. “Intersection Graphs of Quasinormal Subgroups of General Skew Linear Groups”. Hacettepe Journal of Mathematics and Statistics 53/2 (April 2024), 392-404. https://doi.org/10.15672/hujms.1249433.
JAMA
Qui Danh L. Intersection graphs of quasinormal subgroups of general skew linear groups. Hacettepe Journal of Mathematics and Statistics. 2024;53:392–404.
MLA
Qui Danh, Le. “Intersection Graphs of Quasinormal Subgroups of General Skew Linear Groups”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, 2024, pp. 392-04, doi:10.15672/hujms.1249433.
Vancouver
Qui Danh L. Intersection graphs of quasinormal subgroups of general skew linear groups. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):392-404.