A remark on groups without finite quotients

TL;DR

The paper discusses the non-elementary nature of groups without proper finite index subgroups, highlighting examples involving ultrapowers and homomorphisms onto finite cyclic groups.

Contribution

It reveals that the class of nontrivial groups lacking proper finite index subgroups is not elementary, with specific examples involving ultrapowers and homomorphisms.

Findings

Some ultrapowers of certain groups map onto Z/pZ for every prime p

Ultrapowers of simple groups can map onto Z/2Z

The class of such groups is not elementary

Abstract

We notice that the class of nontrivial groups without proper subgroups of finite index is not elementary, because some groups in this class, such as $\mathbb Q*\mathbb Q$, have ultrapowers that map homomorphically onto $\mathbb Z/p\mathbb Z$ for every prime $p$. Also, some ultrapowers of certain simple groups map homomorphically onto $\mathbb Z/2\mathbb Z$.