My supervisors Ian Chiswell and Thomas M¨uller have found a new class of groups of functions defined on intervals of the real line, with multiplication defined by analogy with multiplication in free groups. I have extended this idea to functions defined on a densely ordered abelian group. This doesn’t give rise to a class of groups straight away, but using the idea of exponentiation from a paper by Myasnikov, Remeslennikov and Serbin, I have formed another class of groups, in which each group contains a subgroup isomorphic to one of Chiswell and M¨uller’s groups. After the introduction, the second chapter defines the set that contains the group and describes the multiplication for elements within the set. In chapter three I define exponentiation, which leads on to chapter four, in which I describe how it is used to find my groups. Then in chapter five I describe the structure of the centralisers of certain elements within the groups.