About a normal subgroup

Question

Let $G$ be a group then a subset $H$ of $G$ is called a normal sub group of $G$ if left cosets and right cosets of $ H$ are identical i.e., $Ha=aH$ for all $a\in G$. But, I don't understand how $Ha=Hb$ holds for $a, b\in G$?

Answer 1

For any subgroup $H$ of $G$, $Ha=Hb$ if and only if $ab^{-1} \in H$. The same thing happens with left cosets but with the inverse on the left-hand factor.

So, if you wish to understand examples of how $Ha$ and $Hb$ can be equal, just pick any decent subgroup $H$ of $G$, pick $a \in H$, and note that $Ha=Hb$ whenever $b=ha$ for an $h \in H$. This manufactures plenty of examples for you.