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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

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