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| Paper IPM / M / 2942 |
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| Abstract: | |
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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|=p1α1 p2α2 …pkαk with
p1 < p2 < … < pk where pi's are prime numbers and
αi'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(p1), deg(p2),…, deg(pk)). 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 Ap 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)={29. 39 . 5 . 7, 13} then G ≅ S6(3) or O7 (3), and finally we show that if G is a finite
group such that |G|=29 . 39 . 5 . 7 . 13 and
D(G)=(3,2,2,1,0), then G ≅ S6(3) or O7(3).
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