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Left coset of a subgroup

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Definition

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:

  1. x − 1y 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.

Any element x \in S is termed a coset representative for S.

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).

Equivalence of definitions

For full proof, refer: Equivalence of definitions of left coset

Examples

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.

  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).


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.

Facts

Left congruence

The left cosets of a subgroup are pairwise disjoint, and hence form a partition of the group. The relation of being in the same left coset is an equivalence relation on the group, and this equivalence relation is termed the left congruence induced by the subgroup. Further information: left cosets partition a group

Relation with right coset

Every subset that occurs as a left coset of a subgroup also occurs as a right coset. In fact, the left coset xH occurs as the right coset (xHx − 1)x with xHx − 1 being the new subgroup.

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

Group action

If H \le G, then G acts on the left coset space G / H (i.e., the set of left cosets of H in G) by left multiplication. This action is a transitive action. In fact, the fundamental theorem of group actions shows that any transitive action of a group on a set looks like such an action.

Further information: Group acts on left coset space by left multiplication, Fundamental theorem of group actions

Numerical facts

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.

For full proof, refer: Left cosets are in bijection via left multiplication

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.


References

Facts about Left coset of a subgroupRDF feed
Defined inDummitFoote (?, ?, ?)  +, AlperinBell (?, ?, ?)  +, RobinsonGT (?, ?, ?)  +, RobinsonAA (?, ?, ?)  +, Artin (?, ?, ?)  +, Lang (?, ?, ?)  +, Fraleigh (?, ?, ?)  +, Hungerford (?, ?, ?)  +, Gallian (?, ?, ?)  +, and Herstein (?, ?, ?)  +
Referenced inDummitFoote (?, ?, ?)  +, AlperinBell (?, ?, ?)  +, RobinsonGT (?, ?, ?)  +, RobinsonAA (?, ?, ?)  +, Artin (?, ?, ?)  +, Lang (?, ?, ?)  +, Fraleigh (?, ?, ?)  +, Hungerford (?, ?, ?)  +, Gallian (?, ?, ?)  +, and Herstein (?, ?, ?)  +
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