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Paper   IPM / M / 2942
School of Mathematics
  Title:   A characterization of finite simple groups by the degrees of vertices of their prime graphs
  Author(s):  A. R. Zokayi (joint with A. R. Moghaddamfar and M. R. Darafsheh)
  Status:   Published
  Journal: Algebra Colloq.
  Vol.:  12
  Year:  2005
  Pages:   431-442
  Supported by:  IPM
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 p2α2pkαk 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 GM. Moreover we show that if G is a finite group with OC(G)={29. 39 . 5 . 7, 13} then GS6(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 GS6(3) or O7(3).

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