Quasirecognition by prime graph of U_3(q) where 2 < q =p^{alpha} < 100

Abstract

Let G be a finite group and let Gamma(G) be the prime graphof G. Assume 2 < q = p^{alpha} < 100 . We determine finite groupsG such that Gamma(G) = Gamma(U_3(q)) and prove that if q neq3, 5, 9, 17, then U_3(q) is quasirecognizable by prime graph,i.e., if G is a finite group with the same prime graph as thefinite simple group U_3(q) , then G has a unique non-Abeliancomposition factor isomorphic to U_3(q) . As a consequence of ourresults, we prove that the simple groups U_{3}(8) and U_{3}(11)are 4-recognizable and 2-recognizable by prime graph,respectively. In fact, the group U_{3}(8) is the first examplewhich is a 4-recognizable by prime graph.

Keywords

• Prime graph
• element order
• simple group
• linear group