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Cyclically separated groups

Published online by Cambridge University Press:  17 April 2009

Brian Hartley ,
John C. Lennox  and
Akbar H. Rhemtulla
Brian Hartley
Affiliation:
Department of Mathematics, The University, Manchester M13 9PL, England
John C. Lennox
Affiliation:
Department of Pure Mathematics, University College, Cardiff, Cardiff CFI IXL, United Kingdom
Akbar H. Rhemtulla
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2GI.
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Abstract

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We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L (L ⊲ K ≤ G) is a torsion-free section of G and F(H) is the Fitting subgroup of H then H/F(H) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 26 , Issue 3 , December 1982 , pp. 355 - 384
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Allenby, R.B.J.T. and Tang, C.Y., “The residual finiteness of some one-relator groups”, J. Algebra 71 (1981), 132–140.CrossRefGoogle Scholar
[2]Baumslag, G., Karrass, A. and Solitar, D., “Torsion-free groups and amalgamated products”, Proc. Amer. Math. Soc. 24 (1970), 688–690.CrossRefGoogle Scholar
[3]Берман, С.Д. [Berman, S.D.] “Число неприводимых представлений конечиой групгы над произвольным полем” [The number of irreducible representations of a finite group over an arbitrary field], Dokl. Akad. Nauk SSSR 106 (1956), 767–769.Google ScholarPubMed
[4]Carter, Roger and Hawkes, Trevor, “The F-normalizers of a finite soluble group”, J. Algebra 5 (1967), 175–202.CrossRefGoogle Scholar
[5]Gorenstein, Daniel, Finite groups (Harper and Row, New York, Evanston and London, 1968).Google Scholar
[6]Hall, Philip, Nilpotent groups (Canadian Mathematical Congress Summer Seminar, University of Alberta, 1957). Reprinted: The Edmonton notes on nilpotent groups (Mathematical Notes, Queen Mary-College, London, 1969).Google Scholar
[7]Hartley, B., “On residually finite p-group”, Symposia Mathematica, Vol. XVII, 225–234 (Convegno Sui Gruppi Infinite, INDAM, Roma, 1973. Academic Press, New York and London, 1976).Google Scholar
[8]Hawkes, Trevor, “On metanilpotent Fitting classes”, J. Algebra 63 (1980), 459–483.CrossRefGoogle Scholar
[9]КарГаполов, м.И. [Kargapolov, M.I.], “О раэрешимых ГрУппах конечного ранга” [On solvable groups of finite rank], Algebra i Logika Sem. 1 (1962), no. 5, 37–44.Google ScholarPubMed
[10]Олншанский, А.Ю. [Ol'sanskii˘, A.Ju.], “многообраЭмя финитно аппроксимируемых груп”, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 915–927. English Translation: “Varieties of finitely approximable groups”, Math. USSR Izv. 3 (1969), 867–877.Google ScholarPubMed
[11]Poland, John, “Finite potent groups”, Bull. Austral. Math. Soc. 23 (1981), 111–120.CrossRefGoogle Scholar
[12]Robinson, Derek J.S., Finiteness conditions and generalized soluble groups, Parts 1 and 2 (Ergebnisse der Mathematik und ihrer Grenzgebie, 62, 63. Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[13]Suzuki, M., “Finite groups with nilpotent centralizers”, Trans. Amer. Math. Soc. 99 (1961), 425–470.CrossRefGoogle Scholar