A Characterization of F₄(q) where q=2ⁿ (n > 1)
by
A. Iranmanesh
Department of Mathematics, Tarbiat Modarres University, P.O.Box; 14115-137, Tehran, Iran
Coauthors: B. Khosravi (Department of Mathematics, Tarbiat Modarres University, Tehran, Iran)

Let G be a finite group. The prime graph of G is constructed as follows: the set of vertices is \pi(G) consisting of prime numbers dividing the order of the group, two vertices p and q are joined by an edge if and only if G contains an element of order pq . Denote the connected components of the graph by \pi₁, \pi₂, ..., \pi_t . Then |G| can be expressed as a product of m₁, m₂, ..., m_t , where m_i are positive integers with \pi(m_i)=\pi_i . These m_i's are called the order components of G. We use OC(G) to denote the set of order components of G. If G is of even order , then we always assume that 2 is a member of \pi₁ and we call m₂, ..., m_t the odd order components of G. Let \Gamma(G) denote the prime graph of G and let t(\Gamma(G)) denotes the number of order components of G.

In this paper we prove that the simple groups F₄(q) are also uniquely determined by their order components, where q=2ⁿ (n > 1).

Date received: June 22, 2000