Coset

A left coset of a subgroup $H$ in a group $G$ is a subset of $G$ of the form

for some $g\in G$. A right coset is a subset of the form

Left cosets are the equivalence classes of the equivalence relation on $G$ defined by $g\sim g'$ iff $g^{-1}g'\in H$ (equivalently, $gH=g'H$). In particular, the set of left cosets forms a partition of $G$, and the number of distinct left cosets is the index $[G:H]$. If $H$ is a normal subgroup , then $gH=Hg$ for all $g$, and the set of cosets is the underlying set of the quotient group $G/H$.

Examples:

• In the additive group $\mathbb{Z}$ with $H=3\mathbb{Z}$, the cosets are $3\mathbb{Z}$, $1+3\mathbb{Z}$, and $2+3\mathbb{Z}$.
• In $S_3$ with $H=\{e,(12)\}$, the left cosets are $H$, $(13)H=\{(13),(123)\}$, and $(23)H=\{(23),(132)\}$.
• Left and right cosets can differ when $H$ is not normal: with the same $H\le S_3$, one has $(13)H=\{(13),(123)\}$ but $H(13)=\{(13),(132)\}$.