A class of filed theories is examined which have a Lie group as the range of the field. Topologies can be constructed which make the field space a topological group. This permits the wealth of knowledge on topological groups to be applied to the analysis. In particular, it is shown that the group symmetry is necessarily broken by quantum mechanics.

Members of this class have similar structure to boson models which are concerned with guage groups. Some members of the class exhibit topological structures, kinks, which behave like Fermions.

Two particular models of the class are examined in some detail, one simple and intuitive the other more realistic. The Fermi particle characteristics of the simple model are shown to be not typical of the class. The Fermi behaviour of higher dimensional models is of a different nature. Examination of Lorentz metrics shows that the more relaistic model is present in General Relativity