The minimal condition on normal subgroups (min-n) is the descending chain condition on the lattice of normal subgroups. De Falco, de Giovanni and Musella (2014) proved that a soluble group satisfying the minimal condition on normal subgroups of infinite rank either has finite rank or satisfies full min-n. The problem asks whether the same conclusion survives when 'soluble' is weakened to 'locally soluble'. The soluble-case arguments lean on classical structure theory — in particular Baer's theorem that every soluble min-n group is a torsion group — which fails in the locally soluble universe, where Heineken and Wilson constructed torsion-free locally soluble groups satisfying min-n. The task is therefore either to construct a locally soluble group of infinite rank whose normal subgroups of infinite rank satisfy the minimal condition while full min-n fails, or to prove that no such group exists.

Background

A group satisfies the minimal condition on normal subgroups (min-n) if every non-empty set of normal subgroups contains a minimal element, i.e., there is no infinite strictly descending chain N1⊳N2⊳N3⊳⋯N_1 \rhd N_2 \rhd N_3 \rhd \cdots of normal subgroups. This is a finiteness condition strictly weaker than the minimal condition on all subgroups: for instance, any group with very few normal subgroups (such as a simple group) satisfies min-n trivially. The Prüfer rank (or special rank) of a group is the smallest integer rr such that every finitely generated subgroup can be generated by rr elements; a group has infinite rank if no such finite bound exists. A group is soluble if its derived series — the sequence obtained by repeatedly passing to the subgroup generated by all commutators — reaches the identity after finitely many steps. The theory of soluble groups with min-n is classical: Baer proved that every soluble group satisfying min-n is a torsion group (all elements have finite order), and the structure of such groups was developed by McDougall and by Hartley, who in particular constructed uncountable soluble groups satisfying min-n. The situation changes drastically in the wider universe of locally soluble groups (groups in which every finitely generated subgroup is soluble, a strictly larger class than soluble groups): Heineken and Wilson proved that Baer's periodicity theorem fails there, by giving a general method for constructing non-trivial torsion-free, countable, locally soluble groups satisfying min-n whose normal subgroups are even linearly ordered by inclusion; periodic non-soluble locally soluble groups with min-n were known earlier still. Since 2013 the Naples school has pursued a 'large groups' philosophy: impose a property P\mathcal{P} only on subgroups (or normal subgroups) of infinite rank and ask whether P\mathcal{P} then propagates to the whole group — for example, groups all of whose proper subgroups of infinite rank are abelian or permutable, and groups with finitely many conjugacy classes of non-normal subgroups of infinite rank. Within this program, De Falco, de Giovanni and Musella proved the min-n instance for soluble groups: if a soluble group satisfies the minimal condition on normal subgroups of infinite rank (i.e., the poset of normal subgroups having infinite rank satisfies the minimal condition), then the group either has finite rank or satisfies full min-n; in particular, for soluble groups of infinite rank the restricted condition already forces min-n. The disjunctive 'finite rank' branch is unavoidable: a group of finite rank, such as the infinite cyclic group Z\mathbb{Z}, can satisfy the restricted condition vacuously (it has no normal subgroups of infinite rank at all) while failing min-n. Whether the theorem extends from soluble to locally soluble groups is the open question, and the obstruction is precise: Baer's torsion theorem, the engine of the soluble theory, is false for locally soluble groups by the Heineken–Wilson examples.

Problem Statement

Let GG be a locally soluble group of infinite rank which satisfies the minimal condition on normal subgroups of infinite rank; that is, there is no infinite strictly descending chain N1⊳N2⊳N3⊳⋯N_1 \rhd N_2 \rhd N_3 \rhd \cdots of normal subgroups of GG each of which has infinite rank. Must GG satisfy the full minimal condition on normal subgroups (min-n)? Equivalently, in the disjunctive form of the source theorem: if GG is any locally soluble group satisfying the minimal condition on normal subgroups of infinite rank, must GG either have finite rank or satisfy min-n? The soluble case of this statement is the theorem of De Falco, de Giovanni and Musella (2014); the task is to settle the locally soluble case — either by constructing a counterexample (a locally soluble group of infinite rank, satisfying min-n on normal subgroups of infinite rank, that admits an infinite strictly descending chain of normal subgroups, with only finitely many terms of infinite rank) or by proving the implication in the locally soluble universe. No restrictions on periodicity, cardinality, or additional chain conditions may be assumed.

References

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