# 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.