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

Document Type : Research Paper


1 Islamic Azad University

2 Tarbiat Modares University


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


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