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.
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It looks like you can reduce the axioms further to the structure of the division map [ , ] subject to the axioms that [a,b] = [[a,c],[b,c]] and that all [a,a] are equal to a single element.
I don't know how you'd get out of insisting [a,[a,a]]=a. But otherwise it's fine to do this… that's how it is presented in Hall's book actually.https://twitter.com/solifine/status/1212064308311609345 …
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Oh! You're right. I forgot to add the requirement that the division map is surjective. Once you add this, then it follows that [a,[a,a]]= a.
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Thanks! If anyone else is following along, surjectivity gives that a = [x,y] for some x,y. Then we have a = [x,y] = [[x,y],[y,y]] = [a,[y,y]] = [a,[a,a]].
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I linked to a paper in another comment where this is worked out for the case of groupoids.
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