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**Groups with finite conjugacy classes of subnormal subgroups.**
*(English)*
Zbl 0692.20028

Groups in which every subgroup has only finitely many conjugates were characterized by B. H. Neumann [Math. Z. 63, 76-96 (1955; Zbl 0064.25201)] as those in which the centre has finite index. The paper under review deals with the structure of soluble groups for which this restriction is imposed only to subnormal subgroups. A group G is said to be a V-group if each of its subnormal subgroups has a finite number of conjugates, and G is a \(V_ n\)-group if every subnormal subgroup of G has at most n conjugates. Clearly \(V_ 1\)-groups are precisely the T- groups (groups in which normality is a transitive relation) described by D. J. S. Robinson [Proc. Camb. Philos. Soc. 60, 21-38 (1964; Zbl 0123.24901)]. It is shown that, if G is a soluble \(V_ n\)-group, then the index \(| G:\omega (G)|\) is finite and bounded by a function of n, where \(\omega\) (G) is the Wielandt subgroup of G (i.e. the intersection of the normalizers of subnormal subgroups of G). However, the consideration of the infinite dihedral group shows that a similar result does not hold for soluble V-groups. The author also proves that a soluble V-group G contains a normal subgroup of finite index N such that \(N'\) is periodic and every subgroup of \(N'\) is normal in N. It follows that every soluble V-group is metabelian-by-finite, and that a finitely generated soluble V-group is abelian-by-finite. Among other results, it is also shown that a soluble \(V_ n\)-group, which is a p-group for some odd prime \(p>n\), is abelian. All these results can be viewed as analogues of Robinson’s theorems concerning T-groups.

It was also proved by B. H. Neumann that each subgroup of a group G has finite index in its normal closure if and only if the commutator subgroup \(G'\) of G is finite. Restricting the attention to subnormal subgroups, a group G is called a \(T^*\)-group if \(| H^ G:H|\) is finite for each subnormal subgroup H of G. The author proves in particular that every soluble \(T^*\)-group is finite-by-metabelian, and that a finitely generated soluble \(T^*\)-group is abelian-by-finite.

In the proofs of these results an important rôle is played by certain groups of automorphisms of abelian groups, which can be considered as natural generalizations of the group of power automorphisms. Among other results on these automorphism groups, the author shows that, if A is an abelian group and \(\Gamma\) is a group of automorphisms of A such that every subgroup of A has at most n conjugates under the action of \(\Gamma\), then \(\Gamma\) contains a normal subgroup \(\Gamma_ 1\) acting on A as a group of power automorphisms, and such that the index \(| \Gamma:\Gamma_ 1|\) is finite and bounded by a function of n.

It was also proved by B. H. Neumann that each subgroup of a group G has finite index in its normal closure if and only if the commutator subgroup \(G'\) of G is finite. Restricting the attention to subnormal subgroups, a group G is called a \(T^*\)-group if \(| H^ G:H|\) is finite for each subnormal subgroup H of G. The author proves in particular that every soluble \(T^*\)-group is finite-by-metabelian, and that a finitely generated soluble \(T^*\)-group is abelian-by-finite.

In the proofs of these results an important rôle is played by certain groups of automorphisms of abelian groups, which can be considered as natural generalizations of the group of power automorphisms. Among other results on these automorphism groups, the author shows that, if A is an abelian group and \(\Gamma\) is a group of automorphisms of A such that every subgroup of A has at most n conjugates under the action of \(\Gamma\), then \(\Gamma\) contains a normal subgroup \(\Gamma_ 1\) acting on A as a group of power automorphisms, and such that the index \(| \Gamma:\Gamma_ 1|\) is finite and bounded by a function of n.

Reviewer: F.de Giovanni

### MSC:

20F24 | FC-groups and their generalizations |

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

20F16 | Solvable groups, supersolvable groups |

20E36 | Automorphisms of infinite groups |

20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |

20F22 | Other classes of groups defined by subgroup chains |

20E07 | Subgroup theorems; subgroup growth |

### Keywords:

transitive normality; finitely many conjugates; soluble groups; subnormal subgroups; \(V_ 1\)-groups; T-groups; soluble \(V_ n\)-group; Wielandt subgroup; soluble V-groups; subgroup of finite index; soluble \(T^*\)- group; finite-by-metabelian; abelian-by-finite; group of power automorphisms; automorphism groups
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\textit{C. Casolo}, Rend. Semin. Mat. Univ. Padova 81, 107--149 (1989; Zbl 0692.20028)

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