Right coset of a subgroup
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition
Definition with symbols
Let be a subgroup of a group
. Then, a right coset of
is a nonempty subset
satisfying the following equivalent properties:
-
is in
for any
and
in
, and for any fixed
, the map
is a surjection from
to
- There exists a
in
such that
- For any
in
,
-
is an orbit in
under the action of
on
by left multiplication
Facts
Right congruence
The right cosets of a subgroup are pairwise disjoint, and hence form a partition of the group. The relation of being in the same right coset is an equivalence relation on the group, and this equivalence relation is termed the right congruence induced by the subgroup.
Relation with left coset
Every subset that occurs as a right coset of a subgroup also occurs as a left coset. In fact, the right coset occurs as the left coset
with
being the new subgroup.
Condition for a left coset to also be a right coset
For a given group , a subset is both a left coset of
and a right coset of
if it is of the form
where
is in the normalizer of
. In other words, the normalizer of a subgroup can be defined as the union of those subsets that are both left and right cosets of
.
Numerical facts
Size of each right coset
Let be a subgroup of
and
be any element of
. Then, the map sending
in
to
is a bijection from
to
.
Number of right cosets
The number of right cosets of a subgroup is termed the index of that subgroup.
Since all right cosets have the same size as the subgroup, we have a formula for the index of the subgroup when the whole group is finite: it is the ratio of the order of the group to the order of the subgroup.
This incidentally also proves Lagrange's theorem -- the order of any subgroup of a finite group divides the order of the whole group.
Natural isomorphism of left cosets with right cosets
There is a natural bijection between the set of left cosets of a subgroup and the set of right cosets of that subgroup. This bijection arises from the natural antiautomorphism of a group defined by the map sending each element to its inverse. Further information: Left and right coset spaces are naturally isomorphic