Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 1145-1153, 2017


On the intersection graph of a finite group

Hossein Shahsavari, Behrooz Khosravi

Received August 20, 2016.   First published October 18, 2017.

Abstract:  For a finite group $G$, the intersection graph of $G$ which is denoted by $\Gamma(G)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H\cap K\neq1$. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of ${\rm Aut}(\Gamma(G))$.
Keywords:  intersection graph; regular graph; simple group; automorphism group
Classification MSC:  05C25, 20E32
DOI:  10.21136/CMJ.2017.0446-16

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References:
[1] S. Akbari, F. Heydari, M. Maghasedi: The intersection graph of a group. J. Algebra Appl. 14 (2015), Article ID 1550065, 9 pages. DOI 10.1142/S0219498815500656 | MR 3323326 | Zbl 1309.05090
[2] A. Ballester-Bolinches, R. Esteban-Romero, D. J. S. Robinson: On finite minimal non-nilpotent groups. Proc. Am. Math. Soc. 133 (2005), 3455-3462. DOI 10.1090/S0002-9939-05-07996-7 | MR 2163579 | Zbl 1082.20006
[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford (1985). MR 0827219 | Zbl 0568.20001
[4] B. Csákány, G. Pollák: The graph of subgroups of a finite group. Czech. Math. J. 19 (1969), 241-247. (In Russian.) MR 0249328 | Zbl 0218.20019
[5] M. Hall, Jr.: The Theory of Groups. Chelsea Publishing Company, New York (1976). MR 0414669 | Zbl 0919.20001
[6] S. Kayacan, E. Yaraneri: Abelian groups with isomorphic intersection graphs. Acta Math. Hung. 146 (2015), 107-127. DOI 10.1007/s10474-015-0486-9 | MR 3348183 | Zbl 06659409
[7] S. Kayacan, E. Yaraneri: Finite groups whose intersection graphs are planar. J. Korean Math. Soc. 52 (2015), 81-96. DOI 10.4134/JKMS.2015.52.1.081 | MR 3299371 | Zbl 1314.20016
[8] C. S. H. King: Generation of finite simple groups by an involution and an element of prime order. J. Algebra 478 (2017), 153-173. DOI 10.1016/j.jalgebra.2016.12.031 | MR 3621666 | Zbl 06695595
[9] X. Ma: On the diameter of the intersection graph of a finite simple group. Czech. Math. J. 66 (2016), 365-370. DOI 10.1007/s10587-016-0261-2 | MR 3519607 | Zbl 06604472
[10] D. J. S. Robinson: A Course in the Theory of Groups. Graduate Texts in Mathematics 80, Springer, New York (1996). DOI 10.1007/978-1-4419-8594-1 | MR 1357169 | Zbl 0836.20001
[11] R. Shen: Intersection graphs of subgroups of finite groups. Czech. Math. J. 60 (2010), 945-950. DOI 10.1007/s10587-010-0085-4 | MR 2738958 | Zbl 1208.20022
[12] B. Zelinka: Intersection graphs of finite abelian groups. Czech. Math. J. 25 (1975), 171-174. MR 0372075 | Zbl 0311.05119

Affiliations:   Hossein Shahsavari, Behrooz Khosravi, Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran, e-mail: h.shahsavari13@yahoo.com, khosravibbb@yahoo.com

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