Return to Glossary.

**Formal Definition**

Let $H$ be a subgroup of a group $G$. The subset $aH=\{ah \mid h \in H \}$ of $G$ is the **left coset** of $H$ containing $a$, while $Ha=\{ha \mid h \in H \}$ is the **right coset** of $H$ containing $a$.

**Informal Definition**

A coset is the set that results from operating each element of some subgroup with one of the elements of the associated group.

**Example(s)**

The left cosets of the subgroup $3 \mathbb{Z}$ of $\mathbb {Z}$ are $0 + 3 \mathbb{Z} = 3 \mathbb{Z}, 1 + 3 \mathbb{Z},$ and $2 + 3 \mathbb{Z}$.

Note that $4 + 3 \mathbb{Z} = 1+ 3 +3 \mathbb{Z} = 1+3 \mathbb{Z}$.

This procedure may be used to show that all left cosets of the subgroup $3 \mathbb{Z}$ of $\mathbb {Z}$ are equal to one of the three cosets shown above. Thus, this listing is complete.

**Non-example(s)**

HW8 Problem 2 gives an example in which the left and right cosets of a subgroup are not the same.

**Additional Comments**

Add any other comments you have about the term here