I want to check which of the following classes are axiomatizable and which are even finitely axiomatizable.
- the class of finite groups
- the class of infinite groups
- the class of groups of order $n$ for some fixed $n$
- the class of torsion groups
- the class of torsion-free groups
- Not axiomatizable due to compactness.
- I think the group axioms plus the sequence of formulae ("there are $n$ distinct elements") should give an axiomatization. If it was finitely axiomatizable then so was the complement (ie all structures that don't describe infinite groups), which seems wrong but I'm not sure how to argue this.
- The group axioms plus "there are $n$ distinct elements" plus "there are not $n+1$ distinct elements" should give a finite axiomatization?
- / 5. I'm not sure how to tackle the torsion thing.