On finite simple groups and Kneser graphs.
Let \(G\) be a finite group. Let \(m(G)\) denote the minimal index of a proper subgroup of \(G\). If \(G\) is non-cyclic then \(\sigma(G)\) denotes the least number of proper subgroups of \(G\) whose union is \(G\). If \(G\) is generated by two elements then \(\omega(G)\) denotes the maximal size of subsets \(S\) of \(G\) such that every two distinct elements of \(S\) generate \(G\). Clearly, \(\omega(G)\leq\sigma(G)\) if both invariants are defined. The question arises: does there exist a universal constant \(c\geq 1\) such that if \(G\) is a non-Abelian finite simple group, then \((1-c/m(G)\leq\omega(G))\)? In Theorem 1.1 of the paper, the affirmative answer to this question for projective special linear groups, Suzuki groups and Ree groups is given. Let \(\Gamma(G)\) be the (simple) graph defined on the elements of \(G\) with an edge between two distinct vertices if and only if they generate \(G\). The chromatic number (least number of colors needed to color the vertices of the graph in such way that the endpoints of every edge receive different colors) of the graph \(\Gamma(G)\) is denoted by \(\chi(G)\). Clearly, \(\omega(G)\leq\chi(G)\leq\sigma(G)\) if all three invariants are defined. In Theorem 1.2, if \(G\) is any of the groups \(\text{GL}(n,q)\), \(\text{PGL}(n,q)\), \(\text{SL}(n,q)\), \(\text{PSL}(n,q)\) and \(n\geq 12\), a new lower bound for \(\chi(G)\) is given. In the proofs of Theorems 1.1 and 1.2 certain variants of Kneser graphs appear. In Theorem 1.3, some conditions for the embedding a Kneser graph as induced subgraph to \(\Gamma(S_n)\) or \(\Gamma(A_n)\) are given.
scientific article; zbMATH DE number 5703807
scientific article