# Cosets

If $G$ is a group, $H$ is a subgroup and $a$ be any element in $G$ then the set

is called the right coset generated by $a$ and $H$ is denoted by $Ha$.
Similarly, the set

is called left coset generated by $a$ and $H$ is denoted by $aH$.
Since $eH = He = H$, we see that $H$ itself is a right as well as left coset. Moreover, since $e \in H$, it is evident that $a \in aH$.
If the group operation is addition, we define the right coset of $H$ in $G$ by

Similarly, left coset of $H$ in $G$ is defined by

It must be noticed that cosets are not necessarily subgroups of $G$. They are only special types of complexes which sometimes called residue classes modulo subgroup.
In general $aH \ne Ha$. In case of an abelian group each right coset coincides with the corresponding left coset.

Example: Let $G = \left\{ {a,{a^2},{a^3},{a^4} = 1} \right\}$, and $o\left( G \right) = 4$, $H = \left\{ {1,{a^2}} \right\}$ is a group of $G$. Find all the cosets of $H$ in $G$ and prove that $G$ is equal to union of all these cosets and also establish that any two cosets are either disjoint or identical.

Solution: We have

Thus there are only distinct cosets namely $H$ and $aH$ which is disjoint. Also $H$ is identical with ${a^2}H$ and $aH$ is identical with ${a^3}H$.
Again