The normal complement problem and the structure of the unitary subgroup

Abstract

Let p be an odd prime and G be a finite split metabelian p-group of exponent p. In this article, we obtain a normal complement of G in VðFGÞ, where F is the field with p elements. Further, assume that G ¼ A3C3, where A is a finite abelian p-group and 3 j p 1: If F is any finite field of characteristic p, then we prove that G does not have a normal complement in VðFGÞ and obtain the structure of the unitary subgroup V ðFGÞ: