An example of a statement that is true within a group, but that is unprovable from the group axioms?

Question

We know that:

If our language is the group language $L_G = \{ e, \cdot \}$ and our theory is the three group axioms: $$ (i) \exists e \in G: \forall g \in G: eg = ge = g$$ $$ (ii) \forall g \in G \exists g^{-1} \in G: gg^{-1} = g^{-1}g = e$$ $$ (iii) \forall a,b,c \in G: a(bc) = (ab)c$$ Then every group is a model of that theory.

For a given group of your choice:

1. Can you show me an example of a statement that is true in that model, but that at the same time is also not provable from these axioms? (The model of your choice should be described restrictively by this 3-axioms theory, no other axiom should be added).
2. Which additional axioms would be required to modify the theory in such a way that the statement becomes provable?
3. Can you tell me another model for which that same statement is false?

Answer 1

Consider the model $G = \{e\}$, the trivial group. Then the statement $\forall x\in G : x =e$ is true in the model, but it obviously can't be proved from the axioms since there are groups with more than one element.

Answer 2

Consider an abelian group $A$, for which, by definition, we have for all $a,b\in A$, $$ab=ba.$$ There exist nonabelian groups. Thus the axiom $\forall a,b\in A,\, ab=ba$ is true for some groups but not for others; it cannot, therefore, be derived from the group axioms.