The largest group of invariance for Markov bases and toric ideals

We define the largest group of invariance for a given toric ideal and the associated Markov bases. Reduction by invariance leads to a concise description of an invariant Markov basis and a sampling scheme in terms of the group and a list of representative elements from the orbits of the Markov basis. We also give explicit forms of the largest group of invariance for several standard statistical problems.