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?



Is the equational theory of groups axiomatized by the associative law?
Yes. It suffices to show that any free semigroup embeds in a group.


For this I refer you to [MO question 3235](https://mathoverflow.net/q/3235/4600):



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> Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such situation would be when $G$ is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, $G$ could be solvable).
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