Quasirecognition by prime graph of U_۳(q) where ۲ < q =p^{\alpha} < ۱۰۰

Let G be a finite group and let \Gamma(G) be the prime graph‎ ‎of G‎. ‎Assume ۲ < q = p^{\alpha} < ۱۰۰‎. ‎We determine finite groups‎ ‎G such that \Gamma(G) = \Gamma(U_۳(q)) and prove that if q \neq‎ ‎۳‎, ‎۵‎, ‎۹‎, ‎۱۷‎, ‎then U_۳(q) is quasirecognizable by prime graph‎, ‎i.e‎. ‎if G is a finite group with the same prime graph as the‎ ‎finite simple group U_۳(q)‎, ‎then G has a unique non-Abelian‎ ‎composition factor isomorphic to U_۳(q)‎. ‎As a consequence of our‎ ‎results‎, ‎we prove that the simple groups U_{۳}(۸) and U_{۳}(۱۱)‎ ‎are ۴-recognizable and ۲-recognizable by prime graph‎, ‎respectively‎. ‎In fact‎, ‎the group U_{۳}(۸) is the first example‎ ‎which is a ۴-recognizable by prime graph‎.

Keywords:

prime graph , Element order , simple group , linear group