Prove that this theory is incomplete

Question

Given $\Sigma = \{\forall x \forall y \forall z(x \circ (y \circ z) = (x \circ y) \circ z), \forall x (x \circ e = x), \forall x \exists y (x \circ y = e)\}$ (eg. group theory axioms), I need to prove that $Th(\Sigma)$ is incomplete. I know that I can just find a sentence that is true in some, but not all groups. However I have no idea how to go about doing this. Where do I start?

Answer 1

Hint: Let $\varphi$ be rhe sentence $\forall x\forall y(x=y)$. Then $\varphi$ is true in some groups, and $\lnot\varphi$ is true in some groups.