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Paper IPM / M / 8548  


Abstract:  
Let G be a finite group. The main result of this paper is as
follows: If G is a finite group, such that Γ(G) = Γ(^{2}G_{2}(q)), where q = 3^{2n+1} for some n ≥ 1,
then G has a (unique) nonabelian composition factor isomorphic
to ^{2}G_{2}(q). As a consequence, we prove that if G is a
finite group satisfying G = ^{2}G_{2}(q) and Γ(G) = Γ(^{2}G_{2}(q)) then G ≅ ^{2}G_{2}(q). This enables us
to give new proofs for some theorems; e.g., a conjecture of W. Shi
and J. Bi. Applications of this result are also considered to the
problem of recognition by element orders of finite groups.
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