# Article

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Keywords:
intersection graph; finite simple group; diameter
Summary:
Let \$G\$ be a finite group. The intersection graph \$\Delta _G\$ of \$G\$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of \$G\$, and two distinct vertices \$X\$ and \$Y\$ are adjacent if \$X\cap Y\ne 1\$, where \$1\$ denotes the trivial subgroup of order \$1\$. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound \$28\$. In particular, the intersection graph of a finite non-abelian simple group is connected.
References:
[1] Bosák, J.: The graphs of semigroups. Theory of Graphs and Its Applications Proc. Sympos., Smolenice 1963 Publ. House Czechoslovak Acad. Sci., Praha (1964), 119-125. MR 0173718
[2] Chigira, N., Iiyori, N., Yamaki, H.: Non-abelian Sylow subgroups of finite groups of even order. Invent. Math. 139 (2000), 525-539. DOI 10.1007/s002229900040 | MR 1738059 | Zbl 0961.20017
[3] Csákány, B., Pollák, G.: The graph of subgroups of a finite group. Czech. Math. J. 19 (1969), 241-247 Russian. MR 0249328
[4] Herzog, M., Longobardi, P., Maj, M.: On a graph related to the maximal subgroups of a group. Bull. Aust. Math. Soc. 81 (2010), 317-328. DOI 10.1017/S0004972709000951 | MR 2609113 | Zbl 1196.20037
[5] Huppert, B.: Endliche Gruppen I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134 Springer, Berlin German (1967). MR 0224703 | Zbl 0217.07201
[6] Kayacan, S., Yaraneri, E.: Abelian groups with isomorphic intersection graphs. Acta Math. Hungar. 146 (2015), 107-127. DOI 10.1007/s10474-015-0486-9 | MR 3348183 | Zbl 1374.20047
[7] Kleidman, P. B., Wilson, R. A.: The maximal subgroups of \$J_4\$. Proc. Lond. Math. Soc., III. Ser. 56 (1988), 484-510. DOI 10.1112/plms/s3-56.3.484 | MR 0931511
[8] Kondraťev, A. S.: Prime graph components of finite simple groups. Math. USSR Sb. 67 (1990), Russian 235-247. DOI 10.1070/SM1990v067n01ABEH001363 | MR 1015040 | Zbl 0698.20009
[9] Sawabe, M.: A note on finite simple groups with abelian Sylow \$p\$-subgroups. Tokyo J. Math. 30 (2007), 293-304. DOI 10.3836/tjm/1202136676 | MR 2376509 | Zbl 1151.20008
[10] Shen, R.: 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
[11] Williams, J. S.: Prime graph components of finite groups. J. Algebra 69 (1981), 487-513. DOI 10.1016/0021-8693(81)90218-0 | MR 0617092 | Zbl 0471.20013
[12] Zavarnitsine, A. V.: Finite groups with a five-component prime graph. Sib. Math. J. 54 (2013), 40-46. DOI 10.1134/S0037446613010060 | MR 3089326 | Zbl 1275.20009
[13] Zelinka, B.: Intersection graphs of finite abelian groups. Czech. Math. J. 25 (1975), 171-174. MR 0372075 | Zbl 0311.05119

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