Coset — Definition, Formula & Examples

A coset is the set you get by multiplying every element of a subgroup by a fixed element from the larger group. Left cosets and right cosets partition the group into equal-sized, non-overlapping pieces.

Let $G$ be a group and $H$ a subgroup of $G$ . For any element $g \in G$ , the left coset of $H$ with respect to $g$ is the set $gH = \{gh : h \in H\}$ , and the right coset is $Hg = \{hg : h \in H\}$ . The collection of all distinct left cosets (or right cosets) of $H$ in $G$ forms a partition of $G$ .

Key Formula

gH = \{gh : h \in H\}

Where:

$G$ = The ambient group
$H$ = A subgroup of G
$g$ = A fixed element of G (the coset representative)
$gH$ = The left coset of H determined by g

How It Works

To build a left coset

gH

, pick an element

g

from the group and multiply it on the left with every element of the subgroup

H

. Each coset has exactly

|H|

elements. Two left cosets are either identical or completely disjoint, so they slice the group into equal-sized blocks. The number of distinct cosets, called the index

[G:H]

, satisfies Lagrange's theorem:

|G| = |H| \cdot [G:H]

. When left and right cosets coincide for every

g

, the subgroup

H

is called normal, which is the condition needed to form a quotient group.

Worked Example

Problem: Let

G = \mathbb{Z}_6 = \{0,1,2,3,4,5\}

under addition mod 6, and let

H = \{0,3\}

. Find all distinct left cosets of

H

G

Step 1: Compute the coset

0 + H

by adding 0 to every element of

H

0 + H = \{0+0,\; 0+3\} = \{0, 3\}

Step 2: Compute

1 + H

1 + H = \{1+0,\; 1+3\} = \{1, 4\}

Step 3: Compute

2 + H

2 + H = \{2+0,\; 2+3\} = \{2, 5\}

Step 4: Check the remaining elements.

3 + H = \{3, 0\} = 0 + H

4 + H = \{4, 1\} = 1 + H

, and

5 + H = \{5, 2\} = 2 + H

. No new cosets appear.

Answer: The three distinct left cosets are

\{0,3\}

\{1,4\}

, and

\{2,5\}

. They partition

\mathbb{Z}_6

into 3 blocks of size 2, consistent with Lagrange's theorem:

6 = 2 \times 3

Why It Matters

Cosets are the building blocks of Lagrange's theorem, which explains why the order of a subgroup must divide the order of the group. They also underpin quotient groups, which appear throughout algebra, number theory, and cryptographic protocols like those based on elliptic curves.

Common Mistakes

Mistake: Assuming a coset is itself a subgroup.

Correction: A coset

gH

is a subgroup only when

g \in H

(making it equal to

H

itself). In general, cosets do not contain the identity element and are not closed under the group operation.