Co-maximal Graphs of Subgroups of Groups

Akbari, Saieed ; Miraftab, Babak ; Nikandish, Reza

Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 12-25

Résumé

Let $H$ be a group. The co-maximal graph of subgroups of $H$ , denoted by $\Gamma \left( H \right)$ , is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma \left( H \right)$ if and only if $H\,=\,LK$ . In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of $\Gamma \left( H \right)$ . For instance, we show that if $\Gamma \left( H \right)$ has no isolated vertex, then $\Gamma \left( H \right)$ is connected with diameter at most 3. Also, we characterize all finitely groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma \left( H \right)$ is connected, and moreover, the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

Détail

DOI : 10.4153/CMB-2016-026-0

Mots-clés : 05C25, 05E15, 20D10, 20D15, co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group

Bibliographie

[1] [1] Abdollahi, A., Akbari, S., and Maimani, H. R., Non-commuting graph of a group. J. Algebra 298(2006), 468–492. http://dx.doi.Org/10.1016/j.jalgebra.2006.02.015 Google Scholar

[2] [2] Afkhami, M. and Khashyarmanesh, K., The co-maximal graph of a lattice. Bull. Malays. Math.Sci. Soc 37(2014), 261–269. Google Scholar

[3] [3] Akbari, S., Habibi, M., Majidinya, A., and Manaviyat, R., A note on co-maximal graph of non-commutative rings. Algebras and Representation Theory 16(2013), 303–307. http://dx.doi.Org/1 0.1007/s10468-011 -9309-z Google Scholar

[4] [4] Akbari, S. and Mahdavi-Hezavehi, M., Some special subgroups of GL(D). Algebra Colloq. 5(1998), 361–370. Google Scholar

[5] [5] Akbari, S. and Mahdavi-Hezavehi, M., Normal subgroups of GL(D) are not finitely generated. Proc. Amer. Math. Soc. 128(2000), no. 6, 1627–1632. http://dx.doi.Org/10.1090/S0002-9939-99-05182-5 Google Scholar

[6] [6] Akbari, S., Miraftab, B., and Nikandish, R., A note on co-maximal ideal graph of commutative rings. Ars Combin., to appear. Google Scholar

[7] [7] Akbari, S. and Nikandish, R., Some results on the intersection graphs of ideals of matrix algebras. Linear Multilinear Algebra 62(2014), 195–206. http://dx.doi.Org/10.1080/03081087.2013.769101 Google Scholar

[8] [8] Akbari, S., Nikandish, R., and Nikmehr, M.J., Some results on the intersection graphs of ideals of rings. J. Algebra Appl. 12(2013), no. 4. http://dx.doi.Org/10.1142/S0219498812502003 Google Scholar

[9] [9] Ballester-Bolinches, A. and Guo, X., On complemented subgroups of finite groups. Arch. Math. (Basel) 72(1999), no. 3. 161–166. http://dx.doi.Org/10.1007/s000130050317 Google Scholar

[10] [10] Câlugâreanu, G., Breaz, S., Modoi, C., Pelea, C., and Vâlcan, D., Exercises in abelian group theory. Kluwer Academic Publishers, 2003. Google Scholar

[11] [11] Hall, P., Complemented groups. J. London Math. Soc. 12 (1937), 201–204. http://dx.doi.Org/10.1112/jlms/s1-12.2.201 Google Scholar

[12] [12] Hungerford, T. W., Algebra. Graduate Texts in Mathematics 73, Springer-Verlag, New York, 1980. Google Scholar

[13] [13] Johnson, P. M., A property of factorizable groups. Arch. Math. (Basel) 60(1993), 414–419. http://dx.doi.Org/10.1007/BF01202304 Google Scholar

[14] [14] Kappe, L.-C. and Kirtland, J., Supplementation in groups. Glasgow Math. J. 42(2000), 37–50. http://dx.doi.Org/10.1017/S0017089500010065 Google Scholar

[15] [15] Moconja, S. M. and Petrovic, Z. Z., On the structure of co-maximal graphs of commutative rings with identity. Bull. Aust. Math. Soc. 83(2011), 11–21. http://dx.doi.Org/10.101 7/S0004972710001875 Google Scholar

[16] [16] OFshanskii, A. Y., Geometry of defining relations in groups. Kluwer, 1991. Google Scholar

[17] [17] Scott, W. R., Group theory. Dover Publications, New York, 1987. Google Scholar

[18] [18] Sharma, P. D. and Bhatwadekar, S. M., A note on graphical representation of rings. J. Algebra 176(1995), 124–127. http://dx.doi.Org/10.1006/jabr.1995.1236 Google Scholar

[19] [19] Suzuki, M., Group theory, I. Springer-Verlag, Berlin, 1982. Google Scholar

[20] [20] Wang, H. J., Graphs associated to co-maximal ideals of commutative rings. J. Algebra 320(2008), 2917–2933. http://dx.doi.Org/10.1016/j.jalgebra.2008.06.020 Google Scholar

[21] [21] Wang, H. J., Co-maximal graph of non-commutative rings. Linear Algebra and Appl. 430(2009), 633–641. http://dx.doi.Org/10.1016/j.laa.2008.08.026 Google Scholar

[22] [22] Ye, M. and Wu, T., Co-maximal ideal graphs of commutative rings. J. Algebra Appl. 11(2012), no. 6. http://dx.doi.Org/1 0.1142/S021 9498812 501149 Google Scholar

[23] [23] Zelinka, B., Intersection graphs of finite abelian groups. Czechoslovak Math. J. 25(1975), no. 2, 17–174. Google Scholar