# Article

MSC: 05C25, 20E32
Full entry | Fulltext not available (moving wall 24 months)
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.
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