Fitting subgroups of infinite groups

Question

Fitting subgroup is defined as the subgroup generated by all normal and nilpotent subgroups of a group G. If G is a finite group, we have that Fitting subgroup is nilpotent. If G is infinite not necessarily this occurs. We are looking for examples of infinite groups such that Fitting subgroup is non-nilpotent. Or some criterion to Fitting subgroup to be nilpotent in infinite case. In general, any answer in this direction (possibly with references) is welcome!

Answer 1

There's an obvious criterion: the Fitting subgroup of a group $G$ is nilpotent if and only if there exists $s$ such that every nilpotent normal subgroup $N$ of $G$ is $n$-step nilpotent (i.e., satisfies $N^{s+1}=\{1\}$).

Hence

• to find examples it's enough to produce groups with suitably prescribed nilpotent normal subgroups (as in Derek's comment). Alternatively, the group of upper unipotent (=upper triangular with 1 on diagonal) with entries indexed by $\mathbf{N}$ matrices, over an arbitrary nonzero commutative associative unital ring, has such normal subgroups.
• if for some reason it is known that there is a bound on the step of nilpotency of (normal) subgroups of $G$, then one gets that the Fitting subgroup of $G$ is nilpotent. This applies if $G$ is linear (i.e., isomorphic to a subgroup of $\mathrm{GL}_d(K)$ for some field $K$ and $d<\infty$).

Probably examples of finitely generated groups with non-nilpotent Fitting subgroup are known but this seems much harder to produce.