Left cosets partition a group
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Statement
Verbal statement
The following equivalent statements are true:
- The left cosets of a subgroup in a group partition the group.
- The relation of being in the same left coset is an equivalence relation.
- Every element of the group is in exactly one left coset.
- Any two left cosets of a subgroup either do not intersect, or are equal.
Definitions used
Let be a group, be a subgroup.
For , we say that is in the left coset of if there exists such that .
Proof
The various statements are clearly equivalent. We'll prove (2) here: the relation of being in the same left coset is an equivalence relation.
Reflexivity
Clearly (since is a subgroup). Hence, for any , , so is in the left coset of .
Symmetry
If , for some , then . Since and is a subgroup, . Thus, if is in the left coset of , then is in the left coset of .
Transitivity
If , and , for , and . Since is a subgroup, , so is in the left coset of .
Other proofs
Orbits under a group action
One easy way of seeing that the left cosets partition a group is by viewing the left cosets as orbits of the group under the action of the subgroup by right multiplication.
Left congruence
Another way of viewing the partition of a group into left cosets of a subgroup is in terms of a left congruence. A left congruence on a magma is an equivalence relation with the property that:
The only left congruences on a group are those that arise as partitions in terms of left cosets of a subgroup.
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Proposition 4, Page 80