Coset

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

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License