  In this paper we study groups of positive operators on Banach lattices. If a
certain factorization property holds for the elements of such a group, the
group has a homomorphic image in the isometric positive operators which has the
same invariant ideals as the original group. If the group is compact in the
strong operator topology, it equals a group of isometric positive operators
conjugated by a single central lattice automorphism, provided an additional
technical assumption is satisfied, for which we again have only examples. We
obtain a characterization of positive representations of a group with compact
image in the strong operator topology, and use this for normalized symmetric
Banach sequence spaces to prove an ordered version of the decomposition theorem
for unitary representations of compact groups. Applications concerning spaces
of continuous functions are also considered.
