coset

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Noun

coset (plural cosets)

1. (algebra, group theory) The set that results from applying a group's binary operation with a given fixed element of the group on each element of a given subgroup.
   • 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.

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

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

Translations

• Italian: classe laterale
• Portuguese: classe lateral

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