A Class of Frobenius Groups

Abstract

If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a’ in A, b ≠ 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order.A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transformation x’ = ωx, where ω is a primitive element of K, and by b the transformation x’ = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively.Since G admits a faithful representation on m letters (m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group.