Cosets

Definition

Let $G$ be a group and $H$ be a subgroup of $G$.

• The left coset of $H$ in $G$ determined by an element $g \in G$ is $$gH = \{gh : h \in H\}$$

• The right coset of $H$ in $G$ determined by an element $g \in G$ is $$Hg = \{hg : h \in H\}$$

A coset is therefore the set obtained by multiplying every element of a subgroup by a fixed group element on the left or on the right.

Main Content

1. Left Cosets and Right Cosets

• A left coset is formed by multiplying every element of a subgroup $H$ by a fixed element $g$ from the left. For example, if $H = \{e, a\}$, then $gH = \{ge, ga\} = \{g, ga\}$.
• A right coset is formed by multiplying every element of $H$ by $g$ from the right. In many groups, left and right cosets are different, especially when the group is not abelian. If the group is abelian, then $gH = Hg$ for every $g$.

A useful way to think about cosets is that they are translations of a subgroup. If $H$ is a pattern, then each coset is a shifted copy of that pattern inside the group.

Example:
Let $G = (\mathbb{Z}, +)$ and $H = 4\mathbb{Z} = \{\dots,-8,-4,0,4,8,\dots\}$.
The left and right cosets are the same because addition is commutative. The coset of $1$ is: $$1 + H = \{\dots,-7,-3,1,5,9,\dots\}$$ This is the set of all integers congruent to $1 \pmod{4}$.

2. Properties of Cosets

Equal size

• Every coset of a subgroup has the same number of elements as the subgroup itself. If $H$ has $k$ elements, then each coset $gH$ or $Hg$ also has $k$ elements.

Partition of the group

• The set of all left cosets of $H$ in $G$ divides $G$ into disjoint pieces. Every element of $G$ belongs to exactly one left coset of $H$. The same is true for right cosets.

This partition property is one of the most powerful ideas in group theory. It means the group is completely covered by non-overlapping copies of the subgroup.

Example:
Take $G = \mathbb{Z}$ and $H = 3\mathbb{Z}$. Then the distinct cosets are:

• $0 + H = \{\dots,-6,-3,0,3,6,\dots\}$
• $1 + H = \{\dots,-5,-2,1,4,7,\dots\}$
• $2 + H = \{\dots,-4,-1,2,5,8,\dots\}$

These three cosets partition all integers.

3. Cosets, Index, and Quotient Groups

• The index of a subgroup $H$ in $G$, written $[G:H]$, is the number of distinct left cosets of $H$ in $G$. If $G$ is finite, then $$|G| = [G:H]\cdot |H|$$ This is the statement behind Lagrange’s Theorem.

• Cosets are the building blocks of a quotient group when $H$ is a normal subgroup. The quotient group $G/H$ consists of all cosets of $H$, with the operation $$(gH)(kH) = (gk)H$$ This operation is well-defined only when $H$ is normal in $G$.

Example:
In $\mathbb{Z}$, the quotient group $\mathbb{Z}/4\mathbb{Z}$ has four cosets: $$0+4\mathbb{Z},\; 1+4\mathbb{Z},\; 2+4\mathbb{Z},\; 3+4\mathbb{Z}$$ These correspond to remainders when dividing by 4.

Working / Process

1. Choose a group and a subgroup.

Start with a group $G$ and identify a subgroup $H$. For example, $G = \mathbb{Z}$ and $H = 5\mathbb{Z}$.

2. Form the coset using a group element.

Pick an element $g \in G$ and create the left coset $gH$ or right coset $Hg$. For $g = 2$, $$2 + 5\mathbb{Z} = \{\dots,-8,-3,2,7,12,\dots\}$$

3. Use cosets to analyze structure.

Check how the cosets partition the group, count them to find the index, and determine whether the subgroup is normal if you want to form a quotient group.

Advantages / Applications

• Cosets help classify group elements into equivalence classes, making complex groups easier to study.
• They are essential in proving and understanding Lagrange’s Theorem, which links subgroup size to group size.
• They are used in quotient groups, modular arithmetic, symmetry analysis, and many areas of abstract algebra and number theory.

Summary

• Cosets are formed by multiplying a subgroup by a fixed group element.
• Left and right cosets may differ in non-abelian groups.
• Cosets partition a group into equal-sized, disjoint subsets.
• They are fundamental for subgroup index and quotient group construction.