How to choose ultraproducts to prove that the class of finite ($p$, torsion) groups is not elementary?

Question

I want to use Łoś's theorem to show that finite groups, $p$-groups, and torsion groups do not form elementary classes. Thus, I have to construct the ultraproduct, say, of some finite groups that is not finite. Can you please tell me what groups and ultrafilters I have to choose? Can I restrict myself with ultrafilters over $\omega$? Shall I use principal, or non-principal ultrafilters?

Answer 1

For each $n\in \omega$, let $G_n = \mathbb{Z}/p^n\mathbb{Z}$. Note that each $G_n$ is a finite group, a torsion group, and a $p$-group. Let $U$ be any non-principal ultrafilter on $\omega$. Let $G = \prod_U G_n$ be the ultraproduct of the $G_n$ by $U$. The element represented by $(1,1,1,\dots)$ in $G$ has infinite order. This shows that $G$ is infinite, not torsion, and not a $p$-group.