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Injective modules and soluble groups satisfying the minimal condition for normal subgroups

Published online by Cambridge University Press:  17 April 2009

B. Hartley  and
D. McDougall
B. Hartley
Affiliation:
University of Warwick, Coventry, England; Australian National University, Canberra, ACT.
D. McDougall
Affiliation:
University of Warwick, Coventry, England; Australian National University, Canberra, ACT.
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Abstract

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Let p be a prime and let Q be a centre-by-finite p′-group. It is shown that the ZQ-modules which satisfy the minimal condition on submodules and have p–groups as their underlying additive groups can be classified in terms of the irreducible ZpQ-modules. If such a ZQ-module V is indecomposable it is either the ZpQ-injective hull W of an irreducible ZpQ-module (viewed as a ZQ-module) or is the submodule W[pn] of such a W consisting of the elements ω ∈ W which satisfy pnw = 0.

This classification is used to classify certain abelian-by-nilpotent groups which satisfy Min-n, the minimal condition on normal subgroups. Among the groups to which our classification applies are all quasi-radicable metabelian groups with Min-n, and all metabelian groups which satisfy Min-n and have abelian Sylow p-subgroups for all p.

It is also shown that if Q is any countable locally finite p'-group and V is a ZQ-module whose additive group is a p-group, then V can be embedded in a ZQ-module whose additive group is a minimal divisible group containing that of V. Some applications of this result are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 4 , Issue 1 , February 1971 , pp. 113 - 135
Copyright
Copyright © Australian Mathematical Society 1971

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