Vaughan asks: Is the quasivariety of monoids generated by the groups and the free monoids finitely based? The free monoids seem to be red herrings. The question is equivalent to Is the quasivariety of monoids generated by the groups finitely based? (since every free monoid is a submonoid of a group.) If the answer is known I suspect it would be well-known as a theorem about which monoids can be embedded in groups. I note that a 46-year- old paper by S.I.Adyan is still cited. It establishes just the special case of cancelation monoids given by single defining relations. Vaughn goes on to ask How different is the abelian case? More or fewer axioms? This case is easy. Again, the free commutative monoids are free herrings. Just add to the equational theory of commutative monoids the cancelation principle: xy = xz -> y = z. (Every such monoid appears in the quasivariety since its reflector into the subcat of abelian groups is an embedding.)