Groups with countably many subgroups.

*(English)*Zbl 1342.20030It 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 \(p\).

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 \(p\).

Reviewer: Martyn Dixon (Tuscaloosa)

##### 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 |

##### Keywords:

soluble groups; CMS-groups; countable sets of subgroups; Abelian subgroups; soluble-by-finite groups; minimax groups; group with countably many subgroups
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\textit{G. Cutolo} and \textit{H. Smith}, J. Algebra 448, 399--417 (2016; Zbl 1342.20030)

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