Finite Nonabelian Simple Group Research Articles

A property of finite simple non-abelian groups

It has been proved by Frohlich [1] that the collection of inner automorphisms of a finite simple non-abelian group G generates the group (under pointwise multiplication) of all functions from G into G leaving the identity fixed, and, conversely, that the only finite groups with this property are the simple non-abelian groups, Z2, and [e]. It follows directly that the inner automorphisms and the constant functions of a simple non-abelian group into itself generate the entire group of functions from G into G. The purpose of this note is to prove the following generalization of this theorem.

Finite Groups of Even Order in Which Sylow 2-Groups are Independent

In this paper we will study a class of groups of even order which satisfy the following condition: (TI): two different Sylow 2-groups contain only the identity element in common. There are three series of finite non-abelian simple groups known to satisfy the above condition (TI). Let q denote a power of 2 greater than 2. The linear fractional group in 2 variables over the field of q elements, denoted here by L2(q), satisfies the condition (TI). The projective unitary groups U3(q) provide another series of groups satisfying (TI). The third series consists of groups G(q) defined by the author in [7]. The main result of this paper is the converse of the above statement. We will prove the following theorem.