Left coset of a subgroup

Definition with symbols
Let $$H$$ be a subgroup of a group $$G$$. Then, a left coset of $$H$$ is a nonempty set $$S \subset G$$ satisfying the following equivalent properties:

Any element $$x \in S$$ is termed a coset representative for $$S$$.
 * 1) $$x^{-1}y$$ is in $$H$$ for any $$x$$ and $$y$$ in $$S$$, and for any fixed $$x \in S$$, the map $$y \mapsto x^{-1}y$$ is a surjection from $$S$$ to $$H$$
 * 2) There exists an $$x$$ in $$G$$ such that $$S = xH$$ (here $$xH$$ is the set of all $$xh$$ with $$h \in H$$)
 * 3) For any $$x$$ in $$S$$, $$S = xH$$
 * 4) $$S$$ is one of the orbits in $$G$$ under the right action of $$H$$, i.e. the action of $$H$$ by right multiplication on $$G$$.

Note that $$H$$ is itself a left coset for $$H$$, and we can take as coset representative, any element of $$H$$ (a typical choice would be to take the identity element).

Extreme examples

 * 1) If we consider a group as a subgroup of itself, then there's only one left coset: the subgroup itself.
 * 2) The left cosets of the trivial subgroup in a group are precisely the singleton subsets (i.e. the subsets of size one). In other words, every element forms a coset by itself.

Examples in abelian groups
Note that for abelian groups, since multiplication is commutative, we can drop the left adjective from left cosets. Further, if we use additive notation, the coset of $$g$$ for a subgroup $$H$$ is written as $$g + H$$.


 * 1) In the group of integers under addition, the left cosets of the subgroup of multiples of $$n$$ are the congruence classes mod $$n$$ (i.e. the collections of numbers that leave the same remainder mod $$n$$). For instance, the subgroup of even numbers in the group of integers has two left cosets: the even numbers and odd numbers (coset representatives are 0 and 1 respectively). The subgroup of multiples of 3 has three cosets: the multiples of 3, the numbers that are 1 mod 3, and the numbers that are 2 mod 3. The coset representatives can be taken to be 0,1, and 2 respectively.
 * 2) In the group of rational numbers under addition, the subgroup of integers have, as left cosets, the collections of rational numbers having the same fractional part. The coset representative for a particular coset can be chosen as the unique element in that coset that is in the interval $$[0,1)$$.
 * 3) In a vector space over a field, vector subspaces are examples of subgroups. The cosets of a vector subspace are the "parallel" affine subspaces obtained by translating it. For instance, the two-dimensional vector space $$\R^2$$, which can be identified with the Euclidean plane, the one-dimensional subspaces are lines through the origin. The cosets of any such one-dimensional subspace are precisely the lines parallel to the given line.

Examples in non-Abelian groups

 * 1) In the symmetric group on three elements on elements $$1,2,3$$, any subgroup of order two, say, that obtained by taking the transposition of $$1$$ and $$2$$, has three left cosets. Each coset is described by where it sends the element $$3$$.
 * 2) More generally, in the symmetric group acting on elements $$1,2,3,\ldots,n$$, the subgroup of permutations that fix the element $$n$$ has exactly $$n$$ left cosets: the cosets are parametrized by where they send the element $$n$$.

Size of each left coset
Let $$H$$ be a subgroup of $$G$$ and $$x$$ be any element of $$G$$. Then, the map sending $$g$$ in $$H$$ to $$xg$$ is a bijection from $$H$$ to $$xH$$.

Number of left cosets
The number of left cosets of a subgroup is termed the index of that subgroup.

Since all left 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.