solifine

A group is a set with a distinguished element e and a binary operation [ , ] satisfying: ‣ [a,b] = [[a,c],[b,c]] ‣ [a,e] = a ‣ [a,a] = e for all a,b,c.

Nice! To make this familiar to the unwashed masses, write [a,b] = ab^{-1}.

Can you really derive the associative law from the 3 axioms you listed? That's [[[a,e],[b,e]], [c,e]] = [[a,e], [[b,e],[c,e]]

Yeah, but you had me worried for a minute. First, note that we have: [c,b] = [[c,c],[b,c]] = [e,[b,c]], which I'd like to write as [c,b] = [b,c]⁻¹.

Define a⁻¹ = [e,a] and ab = [a,b⁻¹]. We see this is a good notion of inverse: (a⁻¹)⁻¹=[e,a⁻¹]=[e,[e,a]] = [a,e] = a aa⁻¹=[a,(a⁻¹)⁻¹] = [a,a] = e a⁻¹a = [a⁻¹,a⁻¹] = e