The kernel of π is by definition the set of affine permutations whose underlying permutation is the identity. The window notations of such affine permutations are of the form [ 1 − − a 1 ⋅ ⋅ n , 2 − − a 2 ⋅ ⋅ n , … … , n − − a n ⋅ ⋅ n ] , where ( a 1 , a 2 , … … , a n ) is an integer vector such that a 1 + a 2 + … … + a n = 0 , that is, where ( a 1 , … … , a n ) ∈ ∈ Λ Λ . Geometrically, this kernel consists of the translations, the isometries that shift the entire space V without rotating or reflecting it. In an abuse of notation, the symbol Λ is used in this article for all three of these sets (integer vectors in V, affine permutations with underlying permutation the identity, and translations); in all three settings, the natural group operation turns Λ into an abelian group, generated freely by the n − 1 vectors { ( 1 , − − 1 , 0 , … … , 0 ) , ( 0 , 1 , − − 1 , … … , 0 ) , … … , ( 0 , … … , 0 , 1 , − − 1 ) } .