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

Document Type : Research Paper

Authors

1 Islamic Azad University

2 Tarbiat Modares University

Abstract

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
graph.

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Main Subjects

### History

• Receive Date: 20 September 2011
• Revise Date: 29 February 2012
• Accept Date: 12 April 2012