  Gaussian graphical models have become a well-recognized tool for the analysis
of conditional independencies within a set of continuous random variables. From
an inferential point of view, it is important to realize that they are
composite exponential transformation families. We reveal this structure by
explicitly describing, for any undirected graph, the (maximal) matrix group
acting on the space of concentration matrices in the model. The continuous part
of this group is captured by a poset naturally associated to the graph, while
automorphisms of the graph account for the discrete part of the group. We
compute the dimension of the space of orbits of this group on concentration
matrices, in terms of the combinatorics of the graph; and for dimension zero we
recover the characterization by Letac and Massam of models that are
transformation families. Furthermore, we describe the maximal invariant of this
group on the sample space, and we give a sharp lower bound on the sample size
needed for the existence of equivariant estimators of the concentration matrix.
Finally, we address the issue of robustness of these estimators by computing
upper bounds on finite sample breakdown points.
