Group Theory – Axiomatizability of Some Classes of Groups

Question

I want to check which of the following classes are axiomatizable and which are even finitely axiomatizable.

1. the class of finite groups
2. the class of infinite groups
3. the class of groups of order $n$ for some fixed $n$
4. the class of torsion groups
5. the class of torsion-free groups

Attempts:

1. Not axiomatizable due to compactness.
2. 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.
3. The group axioms plus "there are $n$ distinct elements" plus "there are not $n+1$ distinct elements" should give a finite axiomatization?
4. / 5. I'm not sure how to tackle the torsion thing.

Answer 1

You're correct for 1, 2, and 3. Below are some sketches for how to do the others.

To see why 2 is not finitely axiomatizable, you can take an ultraproduct of $\mathbb{Z}_p$ for $p\in{}\mathbb{N}$ prime. This is an infinite group, so the complement is not closed under ultraproducts, which means that the class of infinite groups is not finitely axiomatizable.

For 4, note that in a torsion group, each element has finite order. Let $C$ be the class of torsion groups and $T$ the theory. There are members of this class with elements of arbitrarily large order (look at $\mathbb{Z}_n$), so you can write sentences that say "there exists an element of order $n$" for each $n\geq{}1$. Then let $T'=T\cup{}\{{}\phi_n\}{}$ for each $n$. Since any finite subset of $T'$ is consistent, by compactness there is a model of $T'$ with an element of infinite order, thus is not a torsion group. So it is not axiomatizable.

For 5, we can do something similar to 2. For torsion-free groups we can include sentences $\phi_n$ that say the only element raised to the $n$th power that equals $e$ is $e$. However it is not finitely axiomatizable since we can take an ultraproduct of groups that have torsion elements and get a torsion free group. For example, take the ultraproduct of $\mathbb{Z}_p$. The structure with universe $\mathbb{Z}_p$ is a torsion-group for each $p$, but the ultraproduct will not be one.