ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

Let G be a finitely generated group. We investigate the graph Gamma M (G), whose vertices are the maximal subgroups of G and where two vertices M(1) and M(2) are joined by an edge whenever M(1) boolean AND M(2) not equal 1. We show that if G is a finite simple group then the graph Gamma M (G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then Gamma M (G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph Gamma M(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph Gamma M (G), then the diameter of Gamma M (G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph Gamma M(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph Gamma M (G), then G must be finite.