  In a group trellis, the sequence of branches that split from the identity
path and merge to the identity path form two normal chains. The Schreier
refinement theorem can be applied to these two normal chains. The refinement of
the two normal chains can be written in the form of a matrix, called the
Schreier matrix form, with rows and columns determined by the two normal
chains.
  Based on the Schreier matrix form, we give an encoder structure for a group
code which is an estimator. The encoder uses the important idea of shortest
length generator sequences previously explained by Forney and Trott. In this
encoder the generator sequences are shown to have an additional property: the
components of the generators are coset representatives in a chain coset
decomposition of the branch group B of the code. Therefore this encoder appears
to be a natural form for a group code encoder. The encoder has a register
implementation which is somewhat different from the classical shift register
structure.
  This form of the encoder can be extended. We find a composition chain of the
branch group B and give an encoder which uses coset representatives in the
composition chain of B. When B is solvable, the generators are constructed
using coset representatives taken from prime cyclic groups.
