TITLE: Is the equational theory of groups axiomatized by the associative law?
QUESTION [8 upvotes]: Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory answer?

REPLY [4 votes]: The title should read " Is the equational theory of (the class of those semigroups which are groups) axiomatized by the associative law?", and the answer given by Bjorn Kjos-Hanssen makes even more sense (and seems more immediate): Take any free group on a set X of generators, and consider its semigroup reduct. The set X generates a subsemigroup S of this reduct, which is free (in semigroups) on X. (If it weren't, the words in X would be identified in the free group.)  So any hope of finding a nontrivial semigroup identity in the equational theory of this class is toast.
Gerhard "Please Pass The Butter Knife" Paseman, 2019.08.31.