Recognizing some finite simple groups by noncommuting graph

摘要

Let G be a nonabelian group. We define the noncommuting graph ?(G) of G as follows: its vertex set is GZ(G), the noncentral elements of G, and two distinct vertices x and y of ?(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ?(G) □? (L) implies G □ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U _4(4), U _4(8), U _4(9), U _4(11), U _4(13), U _4(16), U _4(17) and the projective special linear groups L _9(2), L _(16)(2) are recognizable by noncommuting graph.