I found the following statement with regards to group cosets:
Let $H$ be a subgroup of a group $G$ and suppose that $a, b \in G$.
If $aH = bH$, then $b \in aH$.
My attempt at writing a proof:
Assume $aH = bH$, then for some $h \in H$ we find $b \circ h = b$, thus $b \in bH$.
But $bH = aH$, therefore $b \in aH$ is also true.
My question: Is the reverse also true? Meaning, if $b \in aH$, then $aH = bH$? In other words, can we create an iff statement?
I tried to show that if $b \in aH$, then $aH = bH$, but didn't get very far.
EDIT: Is it possible to show the iff is true without using equivalence relations?