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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.
Additional Comments
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