coset
Contents
English[edit]
Etymology[edit]
co + set; apparently first used 1910 by American mathematician George Abram Miller.
Noun[edit]
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 notes[edit]
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 terms[edit]
Translations[edit]
result of applying group operation with fixed element from the parent group on each element of a subgroup


Further reading[edit]
 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