There is a simple map (technically, a surjective group homomorphism) π from S ~ ~ n onto the finite symmetric group S n . In terms of the combinatorial definition, an affine permutation can be mapped to a permutation by reducing the window entries modulo n to elements of { 1 , 2 , … … , n } , leaving the one-line notation of a permutation. In this article, the image π π ( u ) of an affine permutation u is called the underlying permutation of u.