The usual axioms I've seen for a group are: associativity; existence of two-sided identity; existence of two-sided inverses for all elements.
$$\forall a,b,c\in G: a\left(bc\right)=\left(ab\right)c$$ $$\exists e\in G, \forall a\in G: ae=a=ea$$ $$\forall a\in G \exists a'\in G: aa'=e=a'a$$
I recently came across a different axiomatisation, and there were no proofs of equivalence. They were: associativity; existence of left-identity; existence of left-inverses.
$$\forall a,b,c\in G: a\left(bc\right)=\left(ab\right)c$$ $$\exists e\in G, \forall a\in G: ea=a$$ $$\forall a\in G, \exists a'\in G: a'a=e$$
Are these equivalent? I kind of doubt it, since we have associative semigroups with left but not right identities, but maybe the left-inverses part changes things.
There was a proof that given these axioms, a left-inverse is a right-inverse, and hence that the original inverses axiom is proven, but what about right-identity?
Proof: Let $g\in G$ then $g$ has a left inverse, call it $g'\in G$ and this too has a left inverse, call it $g''\in G$. Then, $g'g=e$, $g''g'=e$ and so $$gg'=egg'=g''g'gg'=g''g'=e$$ so $g'$ is the right-inverse of $g$ also.
from Hot Weekly Questions - Mathematics Stack Exchange
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