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Groups with countably many subgroups. (English) Zbl 1342.20030
It is an easy observation that a group is finite precisely when it has only finitely many subgroups. In the very nice paper under review the authors consider groups $$G$$ in which the set of all subgroups of $$G$$ is countable. The abelian groups with this property are classified in [S. V. Rychkov and A. A. Fomin, “Abelian groups with a countable number of subgroups”, Abelevy Gruppy Moduli 10, 99-105 (1991)]. The current paper is mostly concerned with soluble such groups. A group $$G$$ is termed a CMS-group if $$\mathcal L(G)$$, the set of all subgroups of $$G$$ is countable. Of course, all CMS-groups are countable and also groups with the maximum condition are CMS-groups. The class of CMS-groups is closed under taking subgroups and quotients, but $$C_{p^\infty}\times C_{p^\infty}$$ (which has $$2^{\aleph_0}$$ subgroups) shows that the property is not extension closed.
The authors show (Theorem 2.7) that a soluble-by-finite group is CMS if and only it is minimax and has no (subnormal) sections of type $$C_{p^\infty}\times C_{p^\infty}$$. They also show that locally (soluble-by-finite) CMS-groups are soluble-by-finite (Theorem 2.12). Interestingly also (Theorem 3.1) they give an example of a nilpotent group with uncountably many subgroups, of which just countably many are abelian. On the other hand, if $$G$$ is an infinite soluble-by-finite group then $$|\mathcal L(G)|$$ is either $$\aleph_0$$ or $$2^{|G|}$$. The construction of Theorem 35.2 of A. Yu. Ol’shanskij [Geometry of defining relations in groups. Dordrecht: Kluwer Academic Publishers (1991; Zbl 0732.20019)] provides a simple group of cardinality $$\aleph_1$$ with exactly $$\aleph_1$$ subgroups, all proper subgroups are countable and $$G$$ may be arranged to be a $$p$$-group, for a sufficiently large prime $$p$$.

##### MSC:
 20E15 Chains and lattices of subgroups, subnormal subgroups 20E07 Subgroup theorems; subgroup growth 20F16 Solvable groups, supersolvable groups 20F19 Generalizations of solvable and nilpotent groups 20F50 Periodic groups; locally finite groups
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