Nilpotence of the Fitting subgroup of a soluble group with the minimal condition on normal subgroups

A soluble group may satisfy the minimal condition on normal subgroups (min-n) - no infinite strictly descending chain of normal subgroups - while being far from finite or countable. Robinson (2019) proved that the Fitting subgroup F(G) of any soluble min-n group is hypercentral, and that for metanilpotent such groups (nilpotent length at most 2) F(G) is actually nilpotent, coinciding with the Baer radical. The problem is to decide, at unrestricted generality - no bound on derived length, cardinality, or rank - whether F(G) must be nilpotent for every soluble group with min-n, or whether there exists one (necessarily of derived length at least 3, the range populated by Hartley's uncountable examples) whose Fitting subgroup is hypercentral but not nilpotent.

Background

A group satisfies the minimal condition on normal subgroups, abbreviated min-n, if it has no infinite strictly descending chain of normal subgroups; this is a substantial weakening of the full minimal condition (min), which quantifies over all subgroups. A group satisfying min is a Černikov group: a finite extension of a direct product of finitely many Prüfer groups $C_{p^\infty}$ (the quasicyclic $p$-groups), and these groups are the best-understood infinite min-groups. Soluble groups with min-n form a much wider and subtler class: a classical result recorded in the source paper's introduction shows they are locally finite, metabelian ones are countable, and Hartley (1977) constructed uncountable soluble min-n groups, which by the countability of the metabelian case must have derived length at least 3. The classical theory was built from module-theoretic analyses: Hartley and McDougall (1971) used injective and artinian modules (a module is artinian if it satisfies the descending chain condition on submodules) to decompose the abelian normal layers of metabelian min-n groups, McDougall (1970) established the countability and direct-decomposition theory for metabelian min-n groups, and Silcock (1974) began the study of metanilpotent groups with min-n (a group is metanilpotent if it has a nilpotent normal subgroup with nilpotent quotient, i.e. nilpotent length at most 2). Heineken and Wilson (1974) treated the locally soluble groups with min-n.

The Fitting subgroup $F(G)$ - the subgroup generated by all normal nilpotent subgroups of $G$ - is the central structural tool of finite soluble group theory, where Fitting's theorem guarantees that $F(G)$ is itself nilpotent. For infinite soluble groups this fails in general: a join of normal nilpotent subgroups need not be nilpotent, so the first question for any chain-condition class is whether $F(G)$ is nilpotent, or at least close to it. The relevant relaxations are measured by the transfinite upper central series, defined by iterating $Z_{\alpha+1}/Z_\alpha=Z(G/Z_\alpha)$ through the ordinals: a group is hypercentral if this series reaches the whole group at some stage. Every nilpotent group is hypercentral and every hypercentral group is locally nilpotent, but neither implication reverses; for groups with min-n, nilpotent groups are always Černikov, so nilpotence is a very strong structural conclusion in this setting.

Before 2019 no general theorem on the Fitting subgroup of an arbitrary soluble min-n group was known. Robinson (2019), Theorem 4.1(i), proved that $F(G)$ is hypercentral for every soluble group $G$ with min-n, by constructing an ascending $G$-composition series in the last non-trivial derived term of $G$ and showing that $F(G)$ centralizes every factor. The same paper proves far more in the metanilpotent case: the Baer radical (the join of all subnormal nilpotent subgroups) is nilpotent and coincides with $F(G)$, and $F(G)$ was given an explicit layered structure - a finite-exponent part satisfying the maximal condition on $G$-invariant subgroups, with quotient a direct sum of finitely many $p$-adically irreducible modules (unbounded $p$-groups whose proper submodules are all bounded) for the Černikov quotient $G/F(G)$. What survives untouched is the range of nilpotent length at least 3 - which forces derived length at least 3 - exactly where Hartley's uncountable examples live and where no module-theoretic decomposition over a Černikov quotient is available.

Problem Statement

Let $G$ be a soluble group satisfying the minimal condition on normal subgroups (min-n), and let $F(G)$ denote its Fitting subgroup, the subgroup generated by all normal nilpotent subgroups of $G$. Robinson (2019) proved that $F(G)$ is hypercentral. Decide whether $F(G)$ must in fact be nilpotent, with no restriction on derived length, cardinality, or rank. Since the metanilpotent case (nilpotent length at most 2) is settled affirmatively, the problem takes the following two-sided form: either prove that $F(G)$ is nilpotent for every soluble min-n group $G$ (necessarily extending the argument beyond nilpotent length 2), or exhibit a soluble min-n group $G$ - necessarily of nilpotent length at least 3, and hence of derived length at least 3 - whose Fitting subgroup is hypercentral but not nilpotent. Recall that nilpotent groups satisfying min-n are Černikov groups, so an affirmative answer would give the Fitting subgroup of every soluble min-n group the strongest possible chain-condition structure, while a negative answer would produce the first known examples of soluble min-n groups whose Fitting subgroup is merely hypercentral.

References

1. Derek J. S. Robinson, “On metanilpotent groups satisfying the minimal condition on normal subgroups”, Journal of Group Theory 22 (2019), no. 5, 809–836. DOI: 10.1515/jgth-2018-0210. URL: https://doi.org/10.1515/jgth-2018-0210 (source paper)
2. B. Hartley and D. McDougall, “Injective modules and soluble groups satisfying the minimal condition for normal subgroups”, Bulletin of the Australian Mathematical Society 4 (1971), no. 1, 113–135. DOI: 10.1017/S0004972700046335. URL: https://doi.org/10.1017/S0004972700046335
3. David McDougall, “Soluble groups with the minimum condition for normal subgroups”, Mathematische Zeitschrift 118 (1970), no. 3, 157–167. DOI: 10.1007/BF01113337. URL: https://doi.org/10.1007/BF01113337
4. Howard L. Silcock, “Metanilpotent groups satisfying the minimal condition for normal subgroups”, Mathematische Zeitschrift 135 (1974), no. 2, 165–173. DOI: 10.1007/BF01189353. URL: https://doi.org/10.1007/BF01189353
5. H. Heineken and J. S. Wilson, “Locally soluble groups with min-n”, Journal of the Australian Mathematical Society 17 (1974), no. 3, 305–318. DOI: 10.1017/S1446788700017079. URL: https://doi.org/10.1017/S1446788700017079
6. B. Hartley, “Uncountable Artinian Modules and Uncountable Soluble Groups Satisfying Min-n”, Proceedings of the London Mathematical Society (3) 35 (1977), no. 1, 55–75. DOI: 10.1112/plms/s3-35.1.55. URL: https://doi.org/10.1112/plms/s3-35.1.55
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