coset
EnglishEdit
EtymologyEdit
co + set; apparently first used 1910 by American mathematician George Abram Miller.
NounEdit
coset (plural cosets)
 (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 12, 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 (the operation is ), . Then the coset is the set of integers of the form where runs through all elements of .
 1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 12, Dover, 1992, page 597,
Usage notesEdit
Mathematically, given a group with binary operation , element and subgroup , the set , which also defines the left coset if is not assumed to be abelian.
The concept is relevant to the (mathematical) definitions of normal subgroup and quotient group.
Derived termsEdit
TranslationsEdit
result of applying group operation with fixed element from the parent group on each element of a subgroup


Further readingEdit
 Lagrange's theorem (group theory) on Wikipedia.Wikipedia
 Normal subgroup on Wikipedia.Wikipedia
 Quotient group on Wikipedia.Wikipedia
 Coset on Wolfram MathWorld
 Coset in a group on Encyclopedia of Mathematics