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Intersection graphs of quasinormal subgroups of general skew linear groups

Yıl 2024, Cilt: 53 Sayı: 2, 392 - 404, 23.04.2024

Le Qui Danh

https://doi.org/10.15672/hujms.1249433

Öz

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 $\Gamma_{\mathrm{q}}(\mathrm{GL}_n(D))$.

Anahtar Kelimeler

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

Destekleyen Kurum

Vietnam National University HoChiMinh City (VNUHCM)

Proje Numarası

T2022-18-03

Teşekkür

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

Kaynakça

  • [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.

Yıl 2024, Cilt: 53 Sayı: 2, 392 - 404, 23.04.2024

Le Qui Danh

https://doi.org/10.15672/hujms.1249433

Öz

Proje Numarası

T2022-18-03

Kaynakça

  • [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.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Le Qui Danh 0000-0002-1429-8595

Proje Numarası T2022-18-03
Erken Görünüm Tarihi 15 Ağustos 2023
Yayımlanma Tarihi 23 Nisan 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 53 Sayı: 2

Kaynak Göster

APA 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. Nisan 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, sy. 2 (Nisan 2024): 392-404. https://doi.org/10.15672/hujms.1249433.
EndNote Qui Danh L (01 Nisan 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, c. 53, sy. 2, ss. 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 (Nisan 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, c. 53, sy. 2, 2024, ss. 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.

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