Does there exist a verbally simple group, which is not characteristically simple?

Question

Does there exist a verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally simple group is a group without non-trivial proper verbal subgroups.

If such group exists, it has to be infinite, as every finite group is characteristically simple iff it is verbally simple (the proof of this fact can be found here: Does there exist some sort of classification of finite verbally simple groups?). However that proof, relies strongly on mathematical induction by the group order and is thus valid only for finite groups. And I do not know whether such infinite group exists.

Answer 1

Let $S$ be a nonabelian finite simple group, and let $G=\prod_{i=0}^{\infty}S$ be the direct product of infinitely many copies of $S$.

Then $G$ is verbally simple, since every element $(g_i)$ involves only finitely many different $g_i$, and so is in a subgroup $H<G$ isomorphic to $S^n$ for some finite $n$. But $S^n$ is verbally simple, so for any word $w$, either

• $w$ vanishes on $S$, in which case it vanishes on $G$, or
• the values of $w$ on $S^n$ generate $S^n\cong H$, in which case the verbal subgroup of $G$ determined by $w$ is the whole of $G$.

But $G$ is not characteristically simple, since the restricted direct product $\bigoplus_{i=0}^{\infty}S$ is a characteristic subgroup, characterized by the fact that its elements are precisely the elements of $G$ whose centralizers have finite index in $G$.