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. $x^{-1}y$ 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 $G$.

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

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

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 77 (formal definition, along with right coset)
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 5 (definition introduced in paragraph)
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 10
• An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, ^{More info}, Page 51
• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 57, Section 6 (Cosets)
• Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 12
• A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 121 (formal definition, along with right coset)
• Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 38 (definition introduced after Theorem 4.2, which is about left congruence and right congruence; introduced along with right coset)
• Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 132
• Topics in Algebra by I. N. Herstein, ^{More info}, Page 47, Exercise 5 (definition introduced in exercise)