Groups with restriction on their infinite subnormal subgroups.

*(English)*Zbl 0644.20020A group G is called a \(B_ n\)-group if all its subnormal subgroups have defect at most n. If all infinite subnormal subgroups of G have defect at most n, G is called an \(IB_ n\)-group. The structure of soluble \(IB_ n\)-groups was studied by F. de Giovanni and the reviewer [J. Algebra 96, 566-580 (1985; Zbl 0572.20016); Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 4, No.1, 49-56 (1985; Zbl 0605.20026)]. Here the author considers \(IB_ n\)-groups in general, especially in comparison to \(B_ n\)-groups. The following is proved: If G is an infinite \(IB_ n\)- group, there exists an abelian normal subgroup K of G such that K is either finite or a Prüfer group and G/K is a \(B_ n\)-group. Also, every \(IB_ n\)-group is a \(B_{n+1}\)-group.

B\({}_ 1\)-groups are those in which normality is a transitive relation (T-groups), and \(IB_ 1\)-groups are those in which every infinite subnormal subgroup is normal (IT-groups). For IT-groups the following is proved: If G is an infinite IT-group, it is an extension of a T-group by a residually finite T-group, which is either metabelian or abelian-by- finite of finite exponent. Examples are given to show that some statements on IT-groups cannot be strengthened. Finally, it is pointed out that the groups constructed in one of these examples are counterexamples to a theorem by W. Gaschütz [J. Reine Angew. Math. 198, 87-92 (1957; Zbl 0077.250)].

B\({}_ 1\)-groups are those in which normality is a transitive relation (T-groups), and \(IB_ 1\)-groups are those in which every infinite subnormal subgroup is normal (IT-groups). For IT-groups the following is proved: If G is an infinite IT-group, it is an extension of a T-group by a residually finite T-group, which is either metabelian or abelian-by- finite of finite exponent. Examples are given to show that some statements on IT-groups cannot be strengthened. Finally, it is pointed out that the groups constructed in one of these examples are counterexamples to a theorem by W. Gaschütz [J. Reine Angew. Math. 198, 87-92 (1957; Zbl 0077.250)].

Reviewer: S.Franciosi

##### MSC:

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20E07 | Subgroup theorems; subgroup growth |

20E34 | General structure theorems for groups |

##### Keywords:

transitive normality; subnormal subgroups; defect; soluble \(IB_ n\)- groups; infinite \(IB_ n\)-group; abelian normal subgroup; T-groups; IT- groups
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\textit{H. Heineken}, Proc. Edinb. Math. Soc., II. Ser. 31, No. 2, 231--241 (1988; Zbl 0644.20020)

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##### References:

[1] | DOI: 10.1007/BF01580283 · Zbl 0021.21003 |

[2] | DOI: 10.1017/S0305004100037403 |

[3] | DOI: 10.1016/0021-8693(85)90027-4 · Zbl 0572.20016 |

[4] | Gaschütz, J. Reine. Angew. Math. 198 pp 87– (1957) |

[5] | De Giovanni, Boll. Un. Mat. Ital. D 4 pp 49– (1985) |

[6] | Kassens, Die Wielandtlänge endlicher Gruppen (1980) |

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