Left and Right Cosets
students, imagine you are organizing a school dance. You have a big group of students, but some people move together in a special pattern. In abstract algebra, cosets are a way to group elements of a group by “shifting” a subgroup. This lesson focuses on left cosets and right cosets, which are essential tools in understanding how groups are built and how results like Lagrange’s Theorem work. 🎯
What you will learn
By the end of this lesson, students, you should be able to:
- explain what left and right cosets are,
- tell the difference between them,
- compute examples in familiar groups,
- see how cosets connect to group size and Lagrange’s Theorem,
- use cosets to reason about subgroups and element orders.
Cosets may sound abstract at first, but they are really just a controlled way of moving a subgroup around inside a group. The key idea is simple: take every element of a subgroup and combine it with a fixed element from the group. That creates a new “chunk” of the group. ✨
1. Subgroups as building blocks
To understand cosets, we first need a subgroup. If $G$ is a group and $H$ is a subgroup of $G$, then $H$ is a smaller group sitting inside $G$. It contains the identity element, is closed under the group operation, and contains inverses.
Think of $H$ as a special set of moves in a game. If you can do move $h$ from $H$, then you can also combine moves inside $H$ and stay in $H$. Cosets are what you get when you “offset” this set by another group element.
Suppose $G$ is a group and $a$ is an element of $G$. Then the subgroup $H$ can be shifted by $a$ in two different ways:
- a left coset: $aH = \{ah : h \in H\}$,
- a right coset: $Ha = \{ha : h \in H\}$.
These look very similar, but in general they are not the same. The order matters because group operations may not be commutative.
2. Left cosets: shifting on the left
A left coset of $H$ in $G$ determined by $a$ is the set
$$aH = \{ah : h \in H\}.$$
This means you take each element $h$ in $H$ and multiply it by $a$ on the left. The result is a translated copy of $H$.
Example in $\mathbb{Z}$
Let $G = \mathbb{Z}$ under addition and let $H = 4\mathbb{Z} = \{\dots,-8,-4,0,4,8,\dots\}$. Since the group operation is addition, the left coset of $H$ by $a$ is
$$a + H = \{a + h : h \in H\}.$$
If $a = 1$, then
$$1 + H = \{\dots,-7,-3,1,5,9,\dots\}.$$
If $a = 2$, then
$$2 + H = \{\dots,-6,-2,2,6,10,\dots\}.$$
These cosets are the familiar “classes of numbers with the same remainder mod $4$.” In additive groups, left and right cosets are the same because addition is commutative.
Important facts about left cosets
For any subgroup $H$ and any element $a$ in $G$:
- $aH$ is always nonempty.
- Every element of $aH$ looks like $ah$ for some $h \in H$.
- The element $a$ is always in $aH$, because $ae = a$ where $e$ is the identity in $H$.
That last fact is helpful: every coset contains the “shifting” element itself.
3. Right cosets: shifting on the right
A right coset of $H$ in $G$ determined by $a$ is the set
$$Ha = \{ha : h \in H\}.$$
Now the subgroup elements multiply on the left and the fixed element $a$ sits on the right.
Example in a noncommutative group
To see the difference between left and right cosets, we need a group where order matters. One famous example is the symmetric group $S_3$, the group of all permutations of three objects.
Let
$$H = \{e, (12)\}$$
where $e$ is the identity permutation and $(12)$ swaps $1$ and $2$.
Choose $a = (13)$. Then the left coset is
$$aH = \{(13)e, (13)(12)\} = \{(13), (132)\},$$
and the right coset is
$$Ha = \{e(13), (12)(13)\} = \{(13), (123)\}.$$
These are different sets:
$$aH \neq Ha.$$
This example shows why the words “left” and “right” matter. In a noncommutative group, multiplying on different sides can produce different results. 🔄
4. Why cosets matter: they partition the group
One of the most important facts about cosets is that they divide the group into non-overlapping pieces. In other words, the set of all left cosets of $H$ in $G$ forms a partition of $G$.
That means every element of $G$ belongs to exactly one left coset of $H$.
Why this is true
If two left cosets $aH$ and $bH$ overlap at all, then they are actually the same set. So cosets are either identical or disjoint.
Here is the key idea:
- If $x \in aH \cap bH$, then $x = ah_1 = bh_2$ for some $h_1, h_2 \in H$.
- Rearranging shows that $a$ can be written using $b$ and an element of $H$.
- This implies $aH = bH$.
So you never get partial overlap. That makes cosets very useful for counting.
The same idea holds for right cosets: they also partition the group into disjoint pieces.
5. Index and the size of a subgroup’s footprint
The number of distinct left cosets of $H$ in $G$ is called the index of $H$ in $G$, written
$$[G : H].$$
You can think of index as telling you how many “copies” of $H$ fit into $G$ when the group is split into left cosets.
If $G$ is finite and every coset has the same size as $H$, then
$$|G| = [G : H] \cdot |H|.$$
This is the heart of Lagrange’s Theorem, which says that the order of a subgroup divides the order of the group.
Example
If a finite group $G$ has $12$ elements and a subgroup $H$ has $4$ elements, then the index must be
$$[G : H] = \frac{12}{4} = 3.$$
So there are exactly $3$ left cosets of $H$ in $G$. Each coset has $4$ elements, and together they cover all $12$ elements of $G$.
6. Left and right cosets compared
students, it helps to remember this simple rule:
- In an abelian group, left and right cosets are the same, because $ab = ba$ for all elements.
- In a nonabelian group, they may be different.
A quick comparison
For a subgroup $H$ and element $a$:
- left coset: $aH$,
- right coset: $Ha$.
Even if $aH \neq Ha$, both are still cosets, and both have the same number of elements as $H$ when the group is finite.
This means the choice of left or right does not change the size of a coset, but it can change which elements appear in it.
7. Why every coset has the same size as the subgroup
This fact is crucial for Lagrange’s Theorem.
Take the left coset $aH$. Consider the function
$$f : H \to aH$$
defined by
$$f(h) = ah.$$
This map is one-to-one and onto.
- It is onto because every element of $aH$ has the form $ah$.
- It is one-to-one because if $ah_1 = ah_2$, then multiplying by $a^{-1}$ on the left gives $h_1 = h_2$.
So $|aH| = |H|$.
A similar argument shows that $|Ha| = |H|$ for right cosets.
This is why a finite group can be counted by splitting it into equal-sized cosets. 📦
8. Real-world style example
Suppose a class of students is divided into teams of $3$. If you label one team as $H$, then a “shifted” team can be thought of like a coset. Every team has the same number of students, but the members differ.
In a group, cosets are not just random subsets. They are exact translations of a subgroup. This is why cosets behave so regularly and why they are powerful in proofs.
For example, in modular arithmetic, the subgroup $4\mathbb{Z}$ gives the cosets:
$$0 + 4\mathbb{Z},\quad 1 + 4\mathbb{Z},\quad 2 + 4\mathbb{Z},\quad 3 + 4\mathbb{Z}.$$
These four cosets partition $\mathbb{Z}$ into numbers with the same remainder when divided by $4$.
Conclusion
Left and right cosets are two ways of shifting a subgroup inside a group. A left coset has the form $aH$, while a right coset has the form $Ha$. In abelian groups, these are the same, but in nonabelian groups they may differ. Cosets partition the group, all cosets of a subgroup have the same size as the subgroup, and the number of cosets is the index $[G:H]$. These ideas lead directly to Lagrange’s Theorem, one of the most important results in group theory. students, mastering cosets gives you a strong foundation for understanding subgroup structure, element orders, and the size relationships inside finite groups. ✅
Study Notes
- A left coset of a subgroup $H$ in a group $G$ is $aH = \{ah : h \in H\}$.
- A right coset of $H$ in $G$ is $Ha = \{ha : h \in H\}$.
- In abelian groups, left and right cosets are equal; in nonabelian groups, they may be different.
- Cosets partition the group into disjoint pieces.
- Every coset has the same number of elements as the subgroup $H$.
- The number of left cosets is the index $[G:H]$.
- For finite groups, $|G| = [G:H] \cdot |H|$.
- This counting result is the foundation of Lagrange’s Theorem.
- Cosets help explain how subgroup structure fits inside the whole group.