for every infinite set x we define s(x) as the group of all permutations of x. on its subgroup consisting of all finitely supported permutations there exists a natural group homomorphism signature. however, thanks to an observation of vitali in 1915, we know that this group homomorphism does not extend to s(x). in the talk we extend the signature on the subgroup of s(x) consisting of all piecewise isometric elements (strongly related to the interval exchange transformation group). this allows us to list all of its normal subgroups and gives also informations about an element of the second cohomology group of some groups.