A prime graph of a finite group is defined in the following way: the set of vertices of the graph is the set of prime divisors of the group, and two distinct vertices r and s are adjacent, if there is an element of order rs in the group. In this paper we continue our investigation of the prime graph of a finite simple group started in [1], namely we describe all cocliques of maximal size for all finite simple groups.
It is shown that the condition of nonadjacency of 2 and at least one odd prime in the Gruenberg-Kegel graph of a finite group G under some natural additional conditions suffices to describe the structure of G; in particular, to prove that G has a unique nonabelian composition factor. Applications of this result to the problem of recognition of finite groups by spectrum are also considered.
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