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

Document Type: Research Paper

Authors

1 Islamic Azad University

2 Tarbiat Modares University

Abstract

Let $G $ be a finite group and let $\Gamma(G)$ be the prime graph‎ ‎of G‎. ‎Assume $2 < q = p^{\alpha} < 100$‎. ‎We determine finite groups‎ ‎G such that $\Gamma(G) = \Gamma(U_3(q))$ and prove that if $q \neq‎ ‎3‎, ‎5‎, ‎9‎, ‎17$‎, ‎then $U_3(q)$ is quasirecognizable by prime graph‎, ‎i.e‎. ‎if $G$ is a finite group with the same prime graph as the‎ ‎finite simple group $U_3(q)$‎, ‎then $G$ has a unique non-Abelian‎ ‎composition factor isomorphic to $U_3(q)$‎. ‎As a consequence of our‎ ‎results‎, ‎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 example‎ ‎which is a $4-$recognizable by prime graph‎.

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