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Paper IPM / M / 2942  


Abstract:  
If G is a finite group, we define its prime graph Γ(G),
as follows: its vertices are the primes dividing the order of G
and two distinct vertices p,q are joined by an edge, and denoted
by p ∼ q, if there is an element in G of order pq. Assume
G=p_{1}^{α1} p_{2}^{α2} …p_{k}^{αk} with
p_{1} < p_{2} < … < p_{k} where p_{i}'s are prime numbers and
α_{i}'s are natural number. For p ∈ π(G), let
deg(p)={ q ∈ π(G)p ∼ q}, which we call the degree of
p, and D(G):=(deg(p_{1}), deg(p_{2}),…, deg(p_{k})). In this
paper we prove that, if G is a finite group such that
D(G)=D(M) and G=M, where M is one of the following
simple groups: (1) a sporadic simple group, (2) an alternating
group A_{p} where p and p−2 are primes, (3) some simple groups
of Lie type, then G ≅ M. Moreover we show that if G is a
finite group with OC(G)={2^{9}. 3^{9} . 5 . 7, 13} then G ≅ S_{6}(3) or O_{7} (3), and finally we show that if G is a finite
group such that G=2^{9} . 3^{9} . 5 . 7 . 13 and
D(G)=(3,2,2,1,0), then G ≅ S_{6}(3) or O_{7}(3).
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