Not getting how to prove reverse hypothesis.

Question

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!!

Answer 1

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 $A$.