In the book "A First Course in Abstract Algebra", Edition 7, Fraleigh wrote that
It is possible to give axioms for a group $\left<G,*\right>$ that seem at first glance to be weaker, namely:
- The binary operation $*$ on $G$ is associative.
- There exists a left identity element $e$ in $G$ such that $e*x = x$ for all $x \in G$.
- For each $a \in G$ there exists a left inverse $a'$ in $G$ such that $a' * a = e$
My question is, keep #1 and #2 the same, but can #3 be modified into:
- For each $a \in G$ there exists a right inverse $a'$ in $G$ such that $a * a' = e$
I try to prove it but seems not work, and here is my attempt:
First, I want to show the right inverse is also a left inverse
By #3, $\forall x\in G, \exists x'\in G, x*x'=e$, $ \Rightarrow x'*x*x'=x'*e$
$$ \exists x''\in G, x'*x''=e, \Rightarrow x'*x*x'*x''=x'*e*x''=x'*x'' \Rightarrow x'*x*e=e $$
How to proceed from here? or a counter example?
