A subgroup of a group $G$ is deemed quasinormal (or permutable) if it permutes with all other subgroups of $G$ (i.e., $N$ is quasinormal in $G$ iff $NH = HN$ for all $H \leq G$).
It is known that, if $G$ is finite, then every quasinormal subgroup of $G$ is subnormal (meaning there is a finite chain of type $H_1 \lhd H_2 \lhd … \lhd G$). This post contains an explicit example that the converse is not true.
However, another question that comes up, especially after seeing the Wikipedia article on the subject, which states that “finiteness is essencial”, is:
What is an example of a group (which must be infinite and nonabelian) and a quasinormal subgroup that is not subnormal?
I immediately thought of matrix groups as a possible place to look, but I don’t really know what subgroups to look for.
Thanks in advance!
Examples, which he attributes to Iwasawa, are described by Stonehewer in
S. E. Stonehewer, Permutable subgroups of infinite groups, Math. Z. 125 (1972), 1-16.
(He proves there that quasinormal subgroups are ascendant, and in particular all quasinormal subgroups of finitely generated groups are subnormal.)
I'll copy the description of the examples from the paper.
Let $p$ be a prime, $A$ an abelian group of type $p^\infty$ and let $\alpha$ be a $p$-adic integer, $\alpha \equiv 1 \pmod p$ ($\alpha \equiv 1 \pmod 4 $ if $p=2$). Thus $a \mapsto a^\alpha$, for all $a \in A$, defines an automorphism of $A$. Let $G$ be the split extension of $A$ by the cyclic group $H$ generated by $\alpha$. Then every subgroup of $G$ is permutable in $G$. However, if $\alpha \ne 1$, then $H$ is not subnormal in $G$; and $H$ is core-free, while $H^G=G$ is not residually nilpotent.