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On Finite Essential Extensions of Torsion Free Abelian Groups

Published online by Cambridge University Press:  20 November 2018

K. Benabdallah  and
M. A. Ouldbeddi
K. Benabdallah
Affiliation:
Departement de Mathématiques et de Statistique Université de Montreal C.P. 6128-A Montréal, Quebec H3C3J7 email: benabdal@ere. umontreal. caouldbedd@ere. umontreal. ca
M. A. Ouldbeddi
Affiliation:
Departement de Mathématiques et de Statistique Université de Montreal C.P. 6128-A Montréal, Quebec H3C3J7 email: benabdal@ere. umontreal. caouldbedd@ere. umontreal. ca
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Abstract

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Let A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ xkx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ Ak satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.

Keywords

20K1520K35decompositionfinite extensionrepresentationsubstitutiontorsion freewidth
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 5 , 01 October 1996 , pp. 918 - 929
Copyright
Copyright © Canadian Mathematical Society 1996

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