Products of finite simple groups

Abstract

We call a group an R-group when it has the property that each of its simple, normal subgroups has a solvable outer-automorphism group. In this paper we study R-groups which can be written as the product of two non-Abelian simple groups. We show that if G is an R-group which is the product of two non-Abelian simple groups, then all the proper normal subgroups of G are solvable. We also show that if G is an R-group, G = AB where A, B are non-Abelian simple groups, Z(G) = 1, 1 ≠ N ≠ G, N ◁ G, and ¦ N ¦ = p k for p a prime, then the Sylow p-subgroups of both A and B are non-Abelian. This theorem restricts the possible normal subgroups of a group that is the product of simple groups. Using these and other results we determine all groups, G, which are the product of two non-Abelian simple groups, where the order of the Sylow 2-subgroup of G is less than or equal to 32. We also determine all the groups G = AB, where A, B are simple groups with Sylow 2-subgroups dihedral of order 8. We determine all groups which are the product of two non-Abelian simple groups and which have either Abelian, dihedral, or quasi-dihedral Sylow 2-subgroups.