Let V be a real vector space. The linear subspaces of V are subgroups of the additive group on V. The left (and right) cosets of a fixed 1-dimensional linear subspace L of V are those lines in V that are parallel to L.

H,

(2,3)H = {(2,3),(1,3,2)}, and

(1,3)H = {(1,3),(1,2,3)}.

H,

H(2,3) = {(2,3),(1,2,3)}, and

H(1,3) = {(1,3),(1,3,2)}.