Normal complements and the number two

Abstract

As a means to solving the isomorphism problem many mathematicians have studied the unit group of a group ring. The group G is contained in the group of units. Thus it is beneficial to find out how the group G sits in the unit group. One question that can be asked is: When does G have a normal complement in the unit group of a group ring? In this thesis we will investigate that question by looking at the unit groups of group rings of the form F₂G where G is a group of small order. We will also look at results from two papers by Robert Sandling ([San84b, San89]). In these papers Sandling shows that for modular group algebras of central-elementary-by-abelian p-groups G has a normal complement in the unit group.