×

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.
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
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] R. Baer , Automorphismengruppen von Gruppen mit endlichen Bahnen gleichmassig beschränkter Mächtigkeit , J. Reine Angew. Math. , 262 ( 1973 ), pp. 93 - 119 . MR 325788 | Zbl 0275.20079 · Zbl 0275.20079
[2] C. Casolo , Subgroups of finite index in generalized T-groups , to appear. Numdam | MR 988125 | Zbl 0692.20029 · Zbl 0692.20029
[3] C. Cooper , Power automorphisms of a group , Math. Zeit. , 107 ( 1968 ), pp. 335 - 356 . Article | MR 236253 | Zbl 0169.33801 · Zbl 0169.33801
[4] F. De Giovanni - S. Franciosi , Groups in which every infinite subnormal subgroup is normal , J. Algebra , 96 ( 1985 ), pp. 566 - 580 . MR 810546 | Zbl 0572.20016 · Zbl 0572.20016
[5] L. Fuchs , Infinite Abelian Groups , Academic Press , New York - London ( 1970 - 1973 ). Zbl 0209.05503 · Zbl 0209.05503
[6] H. Heineken , Groups with restrictions on their infinite subnormal subgroups, Proc. Edimburgh Math. Soc. 31 ( 1988 ), pp. 231 - 241 . MR 989756 | Zbl 0644.20020 · Zbl 0644.20020
[7] H. Heineken - J. Lennox , Subgroups of finite index in T-groups , Boll. Un. Mat. It. , ( 6 ) 4-B ( 1985 ), pp. 829 - 841 . MR 831294 | Zbl 0598.20027 · Zbl 0598.20027
[8] F.W. Levi , The ring of automorphisms for which every subgroup of an abelian group is invariant , J. Indian Math. Soc. , 10 ( 1946 ), pp. 29 - 31 . MR 20081 | Zbl 0061.03502 · Zbl 0061.03502
[9] I.D. Macdonald , Some explicit bounds in groups with finite derived groups , Proc. London Math. Soc. , ( 3 ) 11 ( 1961 ), pp. 23 - 56 . MR 124433 | Zbl 0218.20034 · Zbl 0218.20034
[10] B.H. Neumann , Groups covered by permutable subsets , J. London Math. Soc. , 29 ( 1954 ), pp. 236 - 248 . MR 62122 | Zbl 0055.01604 · Zbl 0055.01604
[11] B.H. Neumann , Griups with finite classes of conjugate subgroups , Math. Zeit. , 63 ( 1955 ), pp. 76 - 96 . Article | MR 72137 | Zbl 0064.25201 · Zbl 0064.25201
[12] D.J.S. Robinson , Groups in which normality is a transitive relation , Proc. Cambridge Phil. Soc. , 60 ( 1964 ), pp. 21 - 38 . MR 159885 | Zbl 0123.24901 · Zbl 0123.24901
[13] D.J.S. Robinson , A Course in the Theory of Groups , Springer-Verlag , New York - Heidelberg - Berlin ( 1982 ). MR 648604 | Zbl 0483.20001 · Zbl 0483.20001
[14] D.J.S. Robinson - J. Wiegold , Groups with boundedly finite automorphisms classes , Rend. Sem. Mat. Univ. Padova , 71 ( 1984 ), pp. 273 - 286 . Numdam | MR 769443 | Zbl 0543.20025 · Zbl 0543.20025
[15] W.R. Scott , Group Theory , Prentice-Hall ( 1964 ). MR 167513 | Zbl 0126.04504 · Zbl 0126.04504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.