I have come across an exercise that I cannot solve and honestly, to me, it doesn’t even seem true.
It goes like this:
Let $G$ be a non empty set and $*$ an associative operation on $G$ such that:
- $\exists e\in G:\forall x\in G: x*e = x$ (right identity)
- $\forall x\in G: \exists y\in G: x*y = e$ (right inverse)
Prove that $(G,*)$ is a group. (Hint: start by proving that $g*g = g \implies g = e$).
I tried to solve this in the following way:
$g*g = g \implies (g*g)*r = g*r \implies g*(g*r) = e \implies g*e = e \implies g = e$ where $r$ is a right inverse of g. We can actually see that the statements $g*g = g$ and $g = e$ are equivalent.
After this I tried to prove that the statements implied the existence of a left identity and its equality with $e$, but I had no success so far.
Hope you can help and thank you in advance.