Not getting how to prove reverse hypothesis. This is a theorem from Dummit & Foote text- > Let $G$ be a group acting on the non-empty set $A$.The relation on $A$ defined by > > $a \sim b$ **iff** $a=g.b$ for some $g \in G$ > > is an equivalence relation. I've shown that '$\sim$' is an equivalence relation. But as this theorem is a **bi-implication** statement,i' don't know how to prove **reverse hypothesis** even i'm not getting WHAT is it? Need help in this. Any suggestions are heartly welcome. thank you!!

There is no equivalence (or 'bi-implication'), only the following implication:

> **If** $G$ is a group acting on a non-empty set $A$, **then** the relation $\sim$ on $A$ defined by $$a\sim b\qquad\Leftrightarrow\qquad (\exists g\in G)(a=gb),$$ is an equivalence relation.

The ' **iff** ' in your question only serves to define the relation on $A$.