Questions about cosets: "If $aH\neq Hb$, then $aH\cap Hb=\emptyset$"? Let $H$ be a subgroup of group $G$, and let $a$ and $b$ belong to $G$. Then, it is known that
$$
aH=bH\qquad\text{or}\qquad aH\cap bH=\emptyset 
$$
In other words, $aH\neq bH$ implies $aH\cap bH=\emptyset$. What can we say about the statement

"If $aH\neq Hb$, then $aH\cap Hb=\emptyset$" ?

[EDITED:]
What I think is that when $G$ is Abelian, this can be true since $aH=Ha$ for any $a\in G$. But what if $G$ is non-Abelian? How should I go on?

It is sometimes true and sometimes false.
For example, if $H$ is a normal subgroup of $G$, then it is true.
If $H$ is the subgroup generated by the permutation $(12)$ inside $G=S_3$, the symmetric group of degree $3$, then $(123)H\neq H(132)$, yet $(13)\in(123)H\cap H(132)$
