Quasirecognition by the prime graph of L_3(q) where 3 < q < 100

Abstract

Let $G$ be a finite group. We construct the prime graph of $ G $,<br />which is denoted by $ Gamma(G) $ as follows: the vertex set of this<br />graph is the prime divisors of $ |G| $ and two distinct vertices $ p<br />$ and $ q $ are joined by an edge if and only if $ G $ contains an<br />element of order $ pq $.<br />In this paper, we determine finite groups $ G $ with $ Gamma(G) =<br />Gamma(L_3(q)) $, $2 leq q < 100 $ and prove that if $ q neq 2, 3<br />$, then $L_3(q) $ is quasirecognizable by prime graph, i.e., if $G$<br />is a finite group with the same prime graph as the finite simple<br />group $L_3(q)$, then $G$ has a unique non-Abelian composition factor<br />isomorphic to $L_3(q)$. As a consequence of our results we prove<br />that the simple group $L_{3}(4)$ is recognizable and the simple<br />groups $L_{3}(7)$ and $L_{3}(9)$ are $2-$recognizable by the prime<br />graph.