  For the category of group codes, that generalizes the category of linear
codes over a finite field, and with the generalized notions of direct sums and
ndecomposable group codes, we prove that every MDS non trivial code, every
perfect non trivial code, and every constant weight nondegenerate group code
are indecomposable. We prove that every group code is a direct sum of
indecomposable group codes, and using this result we obtain the automorphism
groups of any group code in terms of its decomposition in indecomposable
components. We conclude with the determination of the structure of decomposable
cyclic group codes.
