In "A First Course in Abstract Algebra", Edition 7, p.43, Fraleigh writes 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 $a' * a = e$
From this one-sided definition, one can prove that the left identity element is also a right identity element, and a left inverse is also a right inverse for the same element. Thus these axioms should not be weaker, since they result in exactly the same structures being called groups. It is conceivable that it might be easier in some cases to check these left axioms that to check our two-sided axioms. Of course, by symmetry it is clear that there are also right axioms for a group.
Does the above few axioms assume that the right identity element exists in the first place?
Consider $a * b = \left|a\right|b$. There are at least two possible solutions for a "right identity element", namely:
$$-1 * x = -1 \implies x = -1$$ $$1 * x = 1 \implies x = 1$$
Even though the group with operation $*$ satisfies axiom 2 of the weaker definition, it seems that $\left<G,*\right>$ cannot be a group, because using the original axioms of a group, there is no identity element $e$ such that $e*a = a * e = a$ for all $a \in G$.
How then do the above axioms "result in exactly the same structures being called groups"?