Showing a group G acts on itself by right multiplication

Question

I want to show that any group $G$ acts on $X = G$ by right multiplication, with action homomorphism $ρ: G → Sym(G)$; $a \rightarrow ρa$ given by $ρa(x) := xa^{-1}$.

I understand how to prove an action, but I don't fully understand what the action is in this case and how to prove it.

Answer 1

If $X = G$, then $X \times G\to G$ defined as $( x,g) = gx^{-1}$, and then check $( x, (y, g)) = ( x, gy^{-1})=(gy^{-1}x^{-1}) =(xy,g)$.