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Paper IPM / M / 8311  


Abstract:  
Let G be a finite group. We denote by Γ(G) the prime
graph of G. 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. We called M a CIT group if
M is of even order and the centralizer of any involution is a
2group. In this paper we determine finite groups G such that
Γ(G) = Γ(M) where M is a CIT simple group. In fact
we prove that if Γ(G) = Γ(PSL(2,p)) and p > 7 is a
Mersenne prime, then G ≅ PSL(2,p). As a consequence of our
results we can give positive answer to a conjecture of W. Shi and
J. Bi.
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