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Paper IPM / M / 8467  


Abstract:  
Let G be a finite group. We define the prime graph
Γ(G) as follows. The vertices of Γ(G) are the
primes dividing the order of G and two distinct
vertices p, q are joined by an edge if there is an
element in G of order pq. Recently M. Hagie
in (Hagie, M. (2003), The prime graph of a sporadic simple group,
Comm. Algebra, 31: 44054424) determined finite groups
G satisfying Γ(G)=Γ(S), where S
is a sporadic simple group. Let p > 3 be a prime number.
In this paper we determine finite groups G such that
Γ(G)=Γ(PSL(2,p)). ALso is prove that if
p > 11 and p\not ≡ 1 (mod 12), then
PSL(2,p) is uniquely determined by its prime graph. As
a consequence of our results we can give positive answer to a
conjecture of W.Shi and J. Bi for the group PSL(2,p).
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