# Coset Decomposition

Let $H$ be a subgroup of group $G$. We know that no right coset of $H$ in $G$ is empty and any two right cosets of $H$ in $G$ are either disjoint or identical. The union of all right cosets of $H$ in $G$ is equal to $G$. Hence the set of all right cosets of $H$ in $G$ gives a partition of $G$.

This partition is called the right coset decomposition of $G$. The procedure to obtain distinct members of this partition is given below:
$H$ itself is a right coset. Now suppose $a \in G$ and $a \notin H$ then $Ha$ will be another distinct right coset. Again let $b$ be another such element that $b \in G$ and $b \notin H$ and also $b \notin Ha$, then $Hb$ will be another distinct right coset. Proceeding in this way, all distinct right cosets of $H$ in $G$ will be obtained.

Thus $G = H \cup Ha \cup Hb \cup Hc \ldots$ where $a,b,c$ are elements of $G$ so chosen that all right cosets are distinct. In the same way, left coset decomposition of $G$ can be obtained.