# Coset in a group

* by a subgroup (from the left)*

A set of elements of of the form

where is some fixed element of . This coset is also called the left coset by in defined by . Every left coset is determined by any of its elements. if and only if . For all the cosets and are either equal or disjoint. Thus, decomposes into pairwise disjoint left cosets by ; this decomposition is called the left decomposition of with respect to . Similarly one defines right cosets (as sets , ) and also the right decomposition of with respect to . These decompositions consist of the same number of cosets (in the infinite case, their cardinalities are equal). This number (cardinality) is called the index of the subgroup in . For normal subgroups, the left and right decompositions coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup.

#### Comments

See also Normal subgroup.

**How to Cite This Entry:**

Coset in a group. O.A. Ivanova (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Coset_in_a_group&oldid=16601