# coset

## English

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### Etymology

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

### 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 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, Springer, 3rd Edition, page 231,
Example 3. Let ${\displaystyle G=\mathbb {Z} }$ (the operation is ${\displaystyle +}$), ${\displaystyle H=2\mathbb {Z} }$. Then the coset ${\displaystyle 1+2\mathbb {Z} }$ is the set of integers of the form ${\displaystyle 1+2k}$ where ${\displaystyle k}$ runs through all elements of ${\displaystyle \mathbb {Z} }$.

#### Usage notes

Mathematically, given a group ${\displaystyle G}$ with binary operation ${\displaystyle \circ }$, element ${\displaystyle g\in G}$ and subgroup ${\displaystyle H\subseteq G}$, the set ${\displaystyle \left\{g\circ h:h\in H\right\}}$, which also defines the left coset if ${\displaystyle G}$ is not assumed to be abelian.

The concept is relevant to the (mathematical) definitions of normal subgroup and quotient group.