There is a canonical way to choose a subgroup of S ~ ~ n that is isomorphic to the finite symmetric group S n . In terms of the algebraic definition, this is the subgroup of S ~ ~ n generated by s 1 , … … , s n − − 1 (excluding the simple reflection s 0 = s n ). Geometrically, this corresponds to the subgroup of transformations that fix the origin, while combinatorially it corresponds to the window notations for which { u ( 1 ) , … … , u ( n ) } = { 1 , 2 , … … , n } (that is, in which the window notation is the one-line notation of a finite permutation).