Coset: Difference between revisions
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* It occurs as a [[left coset]] of some subgroup, or equivalently, its [[left quotient]] is a subgroup | * It occurs as a [[left coset]] of some subgroup, or equivalently, its [[left quotient]] is a subgroup | ||
* It occurs as a [[right coset]] of some subgroup, or equivalently, its [[right quotient]] is a subgroup | * It occurs as a [[right coset]] of some subgroup, or equivalently, its [[right quotient]] is a subgroup | ||
* The translates of the subset under left multiplication by elements of the group | * The translates of the subset under left multiplication by elements of the group are pairwise disjoint and form a partition of the whole group | ||
* The translates of the subset under right multiplication by elements of the group | * The translates of the subset under right multiplication by elements of the group are pairwise disjoint, and form a partition of the whole group. | ||
===Equivalence of definitions=== | |||
{{further|[[equivalence of definitions of coset]]}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 22:30, 10 March 2017
This article defines a property of subsets of groups
View other properties of subsets of groups|View properties of subsets of abelian groups|View subgroup properties
Definition
Symbol-free definition
A subset of a group is said to be a coset if it satisfies the following equivalent conditions:
- It occurs as a left coset of some subgroup, or equivalently, its left quotient is a subgroup
- It occurs as a right coset of some subgroup, or equivalently, its right quotient is a subgroup
- The translates of the subset under left multiplication by elements of the group are pairwise disjoint and form a partition of the whole group
- The translates of the subset under right multiplication by elements of the group are pairwise disjoint, and form a partition of the whole group.
Equivalence of definitions
Further information: equivalence of definitions of coset