In group theory, a coset is a fundamental concept representing a special type of subset formed by "shifting" a subgroup within a larger group. More precisely, a coset is one of the disjoint, equal-size subsets into which a group can be decomposed using one of its subgroups.
When a subgroup $H$ of a group $G$ is considered, it can be used to partition the underlying set of $G$ into these distinct, identically-sized pieces known as cosets. Each coset has the same number of elements (cardinality) as the subgroup $H$ itself.
There are two primary types of cosets:
- Left Cosets: Formed by multiplying elements of the subgroup by a fixed group element from the left.
- Right Cosets: Formed by multiplying elements of the subgroup by a fixed group element from the right.
Defining Cosets
Let $G$ be a group and $H$ be a subgroup of $G$.
-
Left Coset: For any element $g \in G$, the left coset of $H$ with respect to $g$ is denoted by $gH$ and defined as:
$gH = {gh \mid h \in H}$ -
Right Coset: For any element $g \in G$, the right coset of $H$ with respect to $g$ is denoted by $Hg$ and defined as:
$Hg = {hg \mid h \in H}$
It's important to note that while $gH$ and $Hg$ always have the same cardinality (number of elements), they are not necessarily equal. They are equal if and only if the subgroup $H$ is a normal subgroup of $G$.
Key Properties of Cosets
Cosets possess several crucial properties that make them invaluable in understanding group structure:
- Partitioning the Group: The collection of all distinct left (or right) cosets of a subgroup $H$ forms a partition of the group $G$. This means:
- Every element of $G$ belongs to exactly one coset.
- Any two distinct cosets are disjoint (they have no elements in common).
- The union of all cosets equals the entire group $G$.
- Equal Size: All cosets (both left and right) of a given subgroup $H$ have the same cardinality as the subgroup $H$ itself. If $H$ is a finite subgroup, then $|gH| = |Hg| = |H|$ for all $g \in G$.
- Identity Coset: The subgroup $H$ itself is always a coset. Specifically, $eH = H$ (left coset) and $He = H$ (right coset), where $e$ is the identity element of $G$.
- Equivalence Relation: Cosets arise from an equivalence relation. For a subgroup $H$, an equivalence relation $a \sim b$ can be defined as $a^{-1}b \in H$ (for left cosets) or $ab^{-1} \in H$ (for right cosets). The equivalence classes of these relations are precisely the cosets.
The Index of a Subgroup
The number of distinct left (or right) cosets of a subgroup $H$ in a group $G$ is called the index of $H$ in $G$, denoted by $[G:H]$. For finite groups, Lagrange's Theorem states that $|G| = [G:H] \cdot |H|$, which highlights the direct relationship between the order of the group, the order of the subgroup, and the number of cosets.
Example: Cosets in $\mathbb{Z}_6$
Let's consider the group $G = \mathbb{Z}_6 = {0, 1, 2, 3, 4, 5}$ under addition modulo 6.
Let $H = {0, 3}$ be a subgroup of $G$.
We can find the left cosets of $H$ in $\mathbb{Z}_6$:
| Element $g$ | Left Coset $g+H = {g+h \pmod 6 \mid h \in H}$ | Distinct Coset? |
|---|---|---|
| $0$ | $0+H = {0+0, 0+3} = {0, 3}$ | Yes |
| $1$ | $1+H = {1+0, 1+3} = {1, 4}$ | Yes |
| $2$ | $2+H = {2+0, 2+3} = {2, 5}$ | Yes |
| $3$ | $3+H = {3+0, 3+3} = {3, 0} = {0, 3}$ | No (same as $0+H$) |
| $4$ | $4+H = {4+0, 4+3} = {4, 1} = {1, 4}$ | No (same as $1+H$) |
| $5$ | $5+H = {5+0, 5+3} = {5, 2} = {2, 5}$ | No (same as $2+H$) |
In this example, the distinct left cosets are ${0, 3}$, ${1, 4}$, and ${2, 5}$. Notice how these three sets are disjoint and their union is ${0, 1, 2, 3, 4, 5}$, which is the entire group $\mathbb{Z}_6$. Each coset has 2 elements, which is the same as $|H|$.
Since $\mathbb{Z}_6$ is an abelian group (commutative), left cosets will always be equal to right cosets ($g+H = H+g$). This also indicates that $H$ is a normal subgroup of $\mathbb{Z}_6$.
Cosets are fundamental building blocks for constructing quotient groups (or factor groups), which are new groups formed from the cosets themselves, providing a powerful tool for analyzing group structures. To delve deeper into this concept, you can explore resources on cosets in group theory.
