Let $g.a$ denote the left action of group $G$ on a set $A \; \forall g\in G , a\in A$. Let $a.g$ denote the corresponding right action of $G$ on $A$. Then show that both induce the same equivalence relation on A.
Define $\sim, \sim'$ on A as : $\forall a,b \in A$
$a \sim b \iff \exists g\in G \ni a=g.b$
$a \sim' b \iff \exists g \in G \ni a=b.g$
I want to show $a\sim b \iff a\sim' b$
These clearly both are indeed equivalence relation.
Suppose $a\sim b$ then $\exists g \in G \ni a=g.b$ but how do I use this to show $a=b.g' $ for some $g' \in G$?
What am I missing? Help is appreciated.