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Left (or right) cosets of a subgroup form a partition of the ambient group
Cosets Partition a Group: Let be a group and let be a subgroup. Then the set of left cosets forms a partition of (and similarly for right cosets ).
More precisely: every lies in the left coset , and any two left cosets are either equal or disjoint.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
A left coset of a subgroup in a group is a subset of of the form
for some . A right coset is a subset of the form
Left cosets are the equivalence classes of the equivalence relation on defined by iff (equivalently, ). In particular, the set of left cosets forms a partition of , and the number of distinct left cosets is the index . If is a normal subgroup, then for all , and the set of cosets is the underlying set of the quotient group .
A partition of a set is a set of subsets of (called blocks or parts) such that: