Definition of "coset"

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.

1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 1-2, Dover, 1992, page 597

In general, the coset in row x consists of all the elements xh as h runs through the various elements of H.

1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614

Example 3. Let G = Z {\displaystyle G=\mathbb {Z} } (the operation is + {\displaystyle +} ), H = 2 Z {\displaystyle H=2\mathbb {Z} } . Then the coset 1 + 2 Z {\displaystyle 1+2\mathbb {Z} } is the set of integers of the form 1 + 2 k {\displaystyle 1+2k} where k {\displaystyle k} runs through all elements of Z {\displaystyle \mathbb {Z} } .

2009, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer, page 231