Coset

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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.

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