Coset

From Groupprops
Jump to: navigation, search
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 ap artition 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

Relation with other properties

Stronger properties

Related properties

External links

Links to related riders