On the diameter of the intersection graph of a finite simple group

Ma, Xuanlong

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 365-370.

Résumé

Let $G$ be a finite group. The intersection graph ${\mathrm{\Delta }}_G$ 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.

Détail

DOI : 10.1007/s10587-016-0261-2

Classification : 05C25, 20E32
Mots-clés : intersection graph; finite simple group; diameter