Characterization Of Some Finite Simple Groups With Their Order Components

If G is a finite group, we define the prime graph (G) as following: its vertices are the primes dividing the order of G, and two vertices p and q are joined by an edge, if and only if there is an element in G of order pq. We denote the set of all the connected components of graph T{G) by T(G) = {Ï€_i(G), for i = 1,2, .... ,t(G)}, where t(G) is the number of connected components of (G), and if G is of even order we always assume 2 in Ï€₁. We also denote the set of all the primes dividing n by Ï€(n) where n is a natural number. Obviously \G\ can be expressed as a product of m₁,m₂,....., m_t(G), where m_i is a positive integer with Ï€(m_i) = Ï€_i. All m_i are called the order components of G. Let OC{G) = {m₁,m₂,... ,m_t(G)} be the set of order components of G. The order components of non-abelian simple groups having at least two prime graph components have been abtained in [4]Some simple groups are characterized by their order components, In this paper we will prove the following theorems:Theorem 3.1 Let M = ²D_p+1(2), 5 < p ≤ 2^m - 1. If G is a finite group and OC(G) = OC{M), then G≌M.Theorem 3.2 Let M = C_p(2). If G is a finite group and OC(G) = OC(M), then G = M.Theorem 3.3 Let M = ²D_n(3), 9≤n = 2^m +1≠p. If G is a finite group and OC{G) = OC{M), then G≌M.Theorem 3.4 Let M = D_P+1 (3). If G is a finite group and OC(G) = OC{M),...