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In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group . As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to except . Also, we prove that if is an almost simple group related to except and is a finite group such that and , then .
Let be a finite group. The prime graph of is a graph whose vertex set is the set of prime divisors of and two distinct primes and are joined by an edge, whenever contains an element of order . The prime graph of is denoted by . It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if is a finite group such that , where , then has a unique nonabelian composition factor isomorphic to or .
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