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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!

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    $\begingroup$ A direct product of finite groups $G_n$ for $n \ge 0$, where $G_n$ is nilpotent of class $n$ is an infinite group in which the Fitting subgroup (the whole group) is not nilpotent. $\endgroup$ – Derek Holt Nov 6 '19 at 8:16
  • $\begingroup$ "Direct product" in Derek's comment can both be interpreted as "restricted direct product", and "unrestricted direct product". $\endgroup$ – YCor Nov 7 '19 at 15:19
  • $\begingroup$ Thanks for the example, but how can i show that this direct product is not nilpotent? For instance, is true that $\gamma_{m}(G_{1} \times G_{2} \times \dots ) = \gamma_{m}(G_{1}) \times \gamma_{m}(G_{2}) \times \dots $ , where $\gamma_{m}$ is the mth term of lower central series? $\endgroup$ – Inácio Nov 8 '19 at 23:32
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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.

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