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co- +‎ set; apparently first used 1910 by American mathematician George Abram Miller.


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 xaH and ybH; 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, 3rd edition, Springer, 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  .

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


Further readingEdit