  We extend the recent study of the k-body embedded Gaussian ensembles by Benet
et al. (Phys. Rev. Lett. 87 (2001) 101601-1 and Ann. Phys. 292 (2001) 67) and
by Asaga et al. (cond-mat/0107363 and cond-mat/ 0107364). We show that central
results of these papers can be derived directly from the symmetry properties of
both, the many-particle states and the random k-body interaction. We offer new
insight into the structure of the matrix of second moments of the embedded
ensemble, and of the supersymmetry approach. We extend the concept of the
embedded ensemble and define it purely group-theoretically.
