  We show that the commutative closure combined with the iterated shuffle is a
regularity-preserving operation on group languages. In particular, for
commutative group languages, the iterated shuffle is a regularity-preserving
operation. We also give bounds for the size of minimal recognizing automata.
Then, we use these results to deduce that the commutative closure of any
shuffle expression over group languages, i.e., expressions involving shuffle,
iterated shuffle, concatenation, Kleene star and union in any order, starting
with the group languages, always yields a regular language.
