Coset · aiwiki.page

In group theory, a coset is a subset of a group obtained by multiplying every element of a fixed subgroup by the same group element. Multiplication on the left produces a left coset; multiplication on the right produces a right coset. Cosets partition a group into equal-sized subsets, providing the basis for counting subgroup indices and constructing quotient groups. (jmilne.org)

Definition and notation

Let $G$ be a group and $H\leq G$ a subgroup. For $g\in G$ , define

gH=\{gh:h\in H\}, \qquad Hg=\{hg:h\in H\}.

These are respectively the left coset and right coset of $H$ represented by $g$ . The words “left” and “right” indicate the position of $g$ in the product. In a noncommutative group, $gH$ and $Hg$ need not coincide. (math.libretexts.org)

If the operation is written additively, the corresponding notation is

g+H=\{g+h:h\in H\}.

In an abelian group, left and right cosets are identical. A representative is not unique: every element of a coset represents that same coset. (math.libretexts.org)

A coset is a subset, not generally a subgroup. In fact, $gH$ is a subgroup of $G$ exactly when $g\in H$ , in which case $gH=H$ . Otherwise it does not contain the identity element. (math.libretexts.org)

Equality, equivalence classes, and partitions

Two left cosets satisfy

aH=bH \quad\Longleftrightarrow\quad a^{-1}b\in H.

For right cosets, the corresponding criterion is

Ha=Hb \quad\Longleftrightarrow\quad ab^{-1}\in H.

Thus distinct representatives do not necessarily give distinct cosets. Any two left cosets are either equal or disjoint, and the same holds for right cosets. (math.libretexts.org)

The relation

a\sim b \quad\Longleftrightarrow\quad a^{-1}b\in H

is an equivalence relation whose equivalence classes are the left cosets. Every $g\in G$ belongs to $gH$ , so the distinct left cosets form a partition of $G$ . (jmilne.org)

Every coset has the same cardinality as $H$ , because

H\longrightarrow gH,\qquad h\longmapsto gh

is a bijection, with inverse $x\mapsto g^{-1}x$ . This remains true when the sets are infinite; it does not mean that a coset is itself a group under the inherited operation. (jmilne.org)

Examples

Integers and congruence classes

In the additive group of integers, let $H=n\mathbb Z$ , where $n$ is a positive integer. Its cosets are

r+n\mathbb Z=\{r+nk:k\in\mathbb Z\}.

Two integers belong to the same coset exactly when their difference is divisible by $n$ . These are the congruence classes of modular arithmetic. For example, $4\mathbb Z$ has four distinct cosets, represented by $0,1,2,3$ . Each coset is infinite, despite there being only four of them. (math.libretexts.org)

Different left and right cosets

In the symmetric group $S_3$ , take

H=\{e,(12)\},\qquad g=(13).

Using right-to-left composition of permutations,

gH=\{(13),(123)\}, \qquad Hg=\{(13),(132)\}.

The two cosets differ, demonstrating why multiplication order matters. (math.libretexts.org)

Index and Lagrange’s theorem

The index of $H$ in $G$ is the number of distinct left cosets:

[G:H]=|\{gH:g\in G\}|.

It is also the number of right cosets: inversion gives a bijection $gH\mapsto Hg^{-1}$ between the two collections. (jmilne.org)

For finite $G$ , the coset partition proves Lagrange’s theorem:

|G|=[G:H]\,|H|.

Consequently, every subgroup’s order divides the order of $G$ . Applying this to the cyclic subgroup generated by an element shows that the element’s order also divides $|G|$ . The converse is false: a divisor of $|G|$ need not occur as the order of a subgroup. For example, the alternating group $A_4$ has order $12$ but no subgroup of order $6$ . (math.libretexts.org)

For nested subgroups $K\leq H\leq G$ in a finite group, indices multiply:

[G:K]=[G:H][H:K].

For infinite groups, index is still defined by counting cosets, not by ordinary division of infinite cardinalities. (math.libretexts.org)

Normal subgroups and quotient groups

A subgroup $H$ is a normal subgroup when

gH=Hg\qquad\text{for every }g\in G.

Thus normality is a condition on all cosets, not merely on one representative. Every subgroup of an abelian group is normal, and every subgroup of index two is normal. (webpages.csus.edu)

The collection of left cosets is commonly denoted $G/H$ . The proposed multiplication

(aH)(bH)=abH

is independent of the chosen representatives exactly when $H$ is normal. In that case, it makes $G/H$ a quotient group, with identity $H$ and inverse $(gH)^{-1}=g^{-1}H$ . Without normality, the coset set still exists, but this multiplication is not well-defined. (math.libretexts.org)

Group actions and linear algebra

Cosets also describe orbits of a group action. If $G$ acts on a set and

G_x=\{g\in G:g\cdot x=x\}

is the stabilizer of $x$ , then

G/G_x\longrightarrow G\cdot x,\qquad gG_x\longmapsto g\cdot x

is a bijection. For finite groups this yields the orbit–stabilizer formula

|G\cdot x|=[G:G_x]=\frac{|G|}{|G_x|}.

The stabilizer need not be normal: here cosets parameterize an orbit rather than necessarily forming a quotient group. (math.libretexts.org)

In a vector space $V$ , a linear subspace $W$ is also a subgroup under addition. Its cosets are

v+W=\{v+w:w\in W\}.

They form the elements of the quotient vector space $V/W$ , with operations

(v+W)+(u+W)=(v+u)+W,\qquad \lambda(v+W)=\lambda v+W.

The zero element of this quotient is the entire coset $W$ , not a single vector of $V$ . (math.mit.edu)

References

Group Theoryjmilne.org
1: Cosetsmath.libretexts.org
2: Lagrange's Theoremmath.libretexts.org
An Inquiry-Based Approach to Abstract Algebrawebpages.csus.edu
2: Cosets and Factor Groupsmath.libretexts.org
1: Quotient Groupsmath.libretexts.org
1: Groups Acting on Setsmath.libretexts.org
book.dvimath.mit.edu
Multivariable Mathematicspeople.math.harvard.edu