# automorphism group of a given period

Maxim Kontsevich and Don Zagier defined the algebra of periods and conjectured that one can pass from a representation of a given period to another one using only three rules. Assuming this conjecture, can we define the automorphism group of a given period using only these three rules? Can it lead to a method to prove that two periods are equal? Has such a question already been investigated?
Thanks in advance and happy new year.