Left Coset And Right Coset

Formal Definition

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

Informal Definition

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Example(s)

Exhibit the left cosets and the right cosets of the subgroup $3\mathbb{Z}$ of $\mathbb{Z}$.

Left cosets:
$3\mathbb{Z}=\{\dots,-9,-6,-3,0,3,6,9\dots\}$
$1+3\mathbb{Z}=\{\dots,-8,-5,-2,1,4,7,10\dots\}$
$2+3\mathbb{Z}=\{\dots,-7,-4,-1,2,5,8,11\dots\}$
Right cosets:
$3\mathbb{Z}=\{\dots,-9,-6,-3,0,3,6,9\dots\}$
$3\mathbb{Z}+1=\{\dots,-8,-5,-2,1,4,7,10\dots\}$
$3\mathbb{Z}+2=\{\dots,-7,-4,-1,2,5,8,11\dots\}$
Left cosets are same as right cosets because $\mathbb{Z}$ is abelian.

Non-example(s)

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