Coset

1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 1-2, Dover, 1992, page 597, Theorem 10.5. The collection consisting of an invariant subgroup H and all its distinct cosets is itself a group, called the factor group of G, usually denoted by G/H. (Remember that the left and right cosets of an invariant subgroup are identical.) Multiplication of two cosets aH and bH is defined as the set of all distinct products z = xy, with x ∈ aH and y ∈ bH; the identity element of the factor group is the subgroup H itself.

1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614, In general, the coset in row x consists of all the elements xh as h runs through the various elements of H.

Example 3. Let G#61;#92;mathbb#123;Z#125; (the operation is #43;), H#61;2#92;mathbb#123;Z#125;. Then the coset 1#43;2#92;mathbb#123;Z#125; is the set of integers of the form 1#43;2k where k runs through all elements of #92;mathbb#123;Z#125;.