  In this work we construct an automaton for the commutative closure of a given
regular group language. The number of states of the resulting automaton is
bounded by the number of states of the original automaton, raised to the power
of the alphabet size, times the product of the order of the letters, viewed as
permutations of the state set. This gives the asymptotic state bound
$O((n\exp(\sqrt{n\ln n}))^{|\Sigma|})$, if the original regular language is
accepted by an automaton with $n$ states. Depending on the automaton in
question, we label points of $\mathbb N_0^{|\Sigma|}$ by subsets of states and
introduce unary automata which decompose the thus labelled grid. Based on these
constructions, we give a general regularity condition, which is fulfilled for
group languages.
