TITLE: Minimal axioms for a group
QUESTION [14 upvotes]: My group theory is very rusty. If I want to just start with left inverses and left identities, must I link the axioms, or can I leave them independent?
e.g., is it enough to say "there exists at least one $e \in G$ s.t. $ea=a$ for all $a \in G$" and "for each $a$ in $G$, there exists $a^{-1} \in G$ s.t. $a^{-1}a$ is an identity", or must I say "there exists at least one $e \in G$ s.t. ($ea=a$ for all $a \in G$ AND for each $a \in G$ there exists $a^{-1} \in G$ s.t. $a^{-1}a=e$)".
This seems like it must be a FAQ, but I just can't find it. If I don't assume the linkage, I run into problems where I show (for example) that if $a^{-1}a=e_1$, $b=aa^{-1}$, $b^{-1}b=e_2$, then $b=e_2$. That will get me things like $ae_1=a$, but not the general case.

REPLY [2 votes]: It is the linked definition that is formally correct, so it is to be preferred. The problem is that the unlinked definition has become so standardized that we are stuck with it, and the majority of textbooks use it.
In fact, the way it is usually stated admits (at least) two distinct plausible interpretations. Are we assuming the inverse axiom for some $e$ that satisfies the identity axiom, or for all such $e$? Of course it turns out that this makes no difference, but we do not know that at a priori.