Verification of the existence of an inverse in a group In a group $(G,*)$ with neutral element $e$, does the relation $x*y=e$ imply $y*x=e$? i think it is true. Indeed, $y*x=y*(x*y)*x=(y*x)*(y*x)$ hence $(y*x)^n=y*x$ for each $n\geq 2$ which is true only if $y*x=e$. Is this correct? If yes, then to verify the existence of an inverse it suffices to verify only one of the two products is equal to $e$.

I'd say that the step where you go from $(y * x)^n = y * x$ for each $n \geq 2$ to $y * x = e$ needs some reasoning.
However, from $y * x = (y * x) * (y * x)$ you can immediately get $y * x = e$ by the following proposition.
Proposition. Let $g$ be an idempotent element of some group $G$, i.e., $gg = g$. Then $g = e$.
Proof. $g = ge = g(g g^{-1}) = (gg) g^{-1} = g g^{-1} = e$.
