A subgroup $H \le G$ is often too small to describe the whole group, but its copies spread through $G$ in a highly controlled way. The first surprise is that a subgroup does not merely sit inside a group as a smaller object: it partitions the entire group into equal-sized pieces. Those pieces are cosets. They are the mechanism behind congruence classes, quotient groups, the [orbit-stabiliser theorem](/theorems/796), and the counting arguments that make finite group theory work.

The motivating failure is simple. Suppose we want to divide a group $G$ by a subgroup $H \le G$ in the same way that we divide $\mathbb{Z}$ by $n\mathbb{Z}$. For integers, two numbers represent the same remainder modulo $n$ when their difference lies in $n\mathbb{Z}$. In a non-abelian group, there are two possible directions: $g^{-1}x \in H$ and $xg^{-1} \in H$. These need not agree. Cosets are the language that records both directions without pretending that they are the same.

[example: Remainders as Cosets] Let $G = \mathbb{Z}$ under addition and let $H = 3\mathbb{Z} = \{3k : k \in \mathbb{Z}\}$. The three cosets represented by $0,1,2$ are \begin{align*} H &= 0+H = \{3k : k \in \mathbb{Z}\},\\ 1+H &= \{1+3k : k \in \mathbb{Z}\},\\ 2+H &= \{2+3k : k \in \mathbb{Z}\}. \end{align*} Every integer belongs to one of these three sets: by integer division, for each $m \in \mathbb{Z}$ there are $q \in \mathbb{Z}$ and $r \in \{0,1,2\}$ such that \begin{align*} m &= 3q+r, \end{align*} so $m \in r+3\mathbb{Z}$. The three sets are disjoint. For example, if $3r = 1+3s$, then \begin{align*} 1 &= 3r-3s\\ &= 3(r-s), \end{align*} which is impossible because $1$ is not divisible by $3$. Similarly, $3r=2+3s$ would imply $2=3(r-s)$, and $1+3r=2+3s$ would imply $1=3(r-s)$. Thus \begin{align*} \mathbb{Z} &= 3\mathbb{Z} \;\sqcup\; (1+3\mathbb{Z}) \;\sqcup\; (2+3\mathbb{Z}). \end{align*} Finally, ordinary congruence modulo $3$ is exactly equality of these cosets. If $a \equiv b \pmod{3}$, then $b-a=3t$ for some $t \in \mathbb{Z}$, and \begin{align*} b+3k &= a+3t+3k\\ &= a+3(t+k), \end{align*} so $b+3\mathbb{Z} \subseteq a+3\mathbb{Z}$; replacing $t$ by $-t$ gives the reverse inclusion. Conversely, if $a+3\mathbb{Z}=b+3\mathbb{Z}$, then $b \in a+3\mathbb{Z}$, so $b=a+3k$ for some $k \in \mathbb{Z}$, hence $a \equiv b \pmod{3}$. [/example]

The integer example hides two issues that become essential in a general group. First, cosets are sets, not elements. Second, multiplication of cosets is not automatically meaningful unless the subgroup satisfies an additional [symmetry condition](/theorems/1360). The chapter develops these two points from the ground up.

## Definition

The basic operation is to translate a subgroup by a group element. Translation preserves the internal shape of the subgroup, but moves its identity element to another point of the group. Since multiplication may depend on order, the first translation to isolate is multiplication on the left.

[definition: Left Coset] Let $G$ be a group and let $H \le G$. For $g \in G$, the left coset of $H$ with representative $g$ is the subset \begin{align*} gH &= \{gh : h \in H\} \subset G. \end{align*} [/definition]

## Right Cosets and Coset Spaces

The representative $g$ is not part of the coset data in a unique way. Many different elements of $G$ may name the same subset $gH$. To compare this with the other possible order of multiplication in a non-abelian group, we need the corresponding right-handed construction.

[definition: Right Coset] Let $G$ be a group and let $H \le G$. For $g \in G$, the right coset of $H$ with representative $g$ is the subset \begin{align*} Hg &= \{hg : h \in H\} \subset G. \end{align*} [/definition]

In abelian groups, left and right cosets coincide. In non-abelian groups, their difference measures how far $H$ is from being stable under conjugation. The next computation makes the distinction concrete before we package all cosets into a coset space.

[example: Left and Right Cosets in $S_3$] Let $G = S_3$, let $H = \{e,(12)\} \le S_3$, and take $g=(123)$. By the definition of a left coset, \begin{align*} gH &= (123)H\\ &= \{(123)e,\,(123)(12)\}\\ &= \{(123),\,(123)(12)\}. \end{align*} To compute $(123)(12)$, compose right to left: \begin{align*} 1 &\mapsto 2 \mapsto 3,\\ 3 &\mapsto 3 \mapsto 1,\\ 2 &\mapsto 1 \mapsto 2. \end{align*} Thus $(123)(12)$ swaps $1$ and $3$ and fixes $2$, so \begin{align*} (123)(12) &= (13), \end{align*} and therefore \begin{align*} (123)H &= \{(123),(13)\}. \end{align*} By the definition of a right coset, \begin{align*} Hg &= H(123)\\ &= \{e(123),\,(12)(123)\}\\ &= \{(123),\,(12)(123)\}. \end{align*} Again composing right to left, \begin{align*} 1 &\mapsto 2 \mapsto 1,\\ 2 &\mapsto 3 \mapsto 3,\\ 3 &\mapsto 1 \mapsto 2. \end{align*} Thus $(12)(123)$ swaps $2$ and $3$ and fixes $1$, so \begin{align*} (12)(123) &= (23), \end{align*} and therefore \begin{align*} H(123) &= \{(123),(23)\}. \end{align*} Since $(13) \neq (23)$ in $S_3$, the subsets $\{(123),(13)\}$ and $\{(123),(23)\}$ are not equal. Thus the left and right cosets of the same subgroup by the same element can differ in a non-abelian group. [/example]

A single coset records one translated copy of $H$, but many arguments need the whole family of such copies at once. To discuss partitions, indices, and group actions, we need a name for the set of all left cosets.

[definition: Left Coset Space] Let $G$ be a group and let $H \le G$. The left coset space of $H$ in $G$ is the set \begin{align*} G/H &= \{gH : g \in G\}. \end{align*} [/definition]

The notation $G/H$ is suggestive, but at this stage it denotes only a set. Since right cosets can differ from left cosets, the right-handed family needs its own notation rather than being silently identified with $G/H$.

[definition: Right Coset Space] Let $G$ be a group and let $H \le G$. The right coset space of $H$ in $G$ is the set \begin{align*} H \backslash G &= \{Hg : g \in G\}. \end{align*} [/definition]

The next invariant asks how large the coset partition is. Counting translated copies of $H$ is what turns cosets into divisibility statements, so the number of cosets deserves its own name.

[definition: Index of a Subgroup] Let $G$ be a group and let $H \le G$. The index of $H$ in $G$ is the cardinality of the left coset space: \begin{align*} [G : H] &= |G/H|. \end{align*} [/definition]

For groups, the number of left cosets and the number of right cosets agree, so the definition of index is independent of this choice. This fact is part of the structural theory below, not a property that should be assumed from the notation.

## Cosets as Equivalence Classes