Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-06T03:43:28.629Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Azurnaya, Gorô, “On generalized semi-primary rings and Krull-Remak-Schmidt's theorem”, Japan J. Math. 19 (1948), 525–547.Google Scholar
[2]Baer, Reinhold, “Abelian groups that are direct summands of every containing group”, Bull. Amer. Math. Soc. 46 (1940), 800–806.CrossRefGoogle Scholar
[3]Baer, Reinhold, “Nilgruppen”, Math. Z. 62 (1955), 402–437.CrossRefGoogle Scholar
[4]Beer, Reinhold, “Irreducible groups of automorphisms of abelian groups”, Pacific J. Math. 14 (1964), 385–406.Google Scholar
[5]Čarin, V.S., “A remark on the minimal condition for subgroups”, (Russian), Dokl. Akad. Nauk SSSR 66 (1949), 575–576.Google Scholar
[6]Černikov, S.N., “Finiteness conditions in group theory”, (Russian), Uspehi Mat. Nauk 14 (1959), no. 5 (89), 45–96. German translation: “Endlichkeitsbedingungen in der Grupperitheorie”, (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).Google Scholar
[7]Curtis, Charles W.; Reiner, Irving, Representation theory of finite groups and associative algebras (Pure and Applied Mathematics, XI; Interscience, [John Wiley & Sons], New York, London, 1962).Google Scholar
[8]Eckmann, B. und Schopf, A., “Über injektive Moduln”, Arch. Math. 4 (1953), 75–78.CrossRefGoogle Scholar
[9]Gardiner, A.D., Hartley, B. and Tomkinson, N.J., “Saturated formations and Sylow structure in locally finite groups”, J. Algebra (to appear).Google Scholar
[10]Hartley, B., “Sylow subgroups of locally finite groups” Proc. London Math. Soc. (to appear).Google Scholar
[11]Levy, Lawrence, “Torsion-free and divisible modules over non-integral-domains”, Canad. J. Math. 15 (1963), 132–151.Google Scholar
[12]Kovács, L.G. and Newman, M.F., “Direct complementation in groups with operators”, Arch. Math. 13 (1962), 1427–1433.CrossRefGoogle Scholar
[13]McDougall, D., “The subnormal structure of some classes of soluble groups”, J. Austral. Math. Soc. (to appear).Google Scholar
[14]McDougall, D., “Soluble groups with the minimal condition for normal subgroups”, Math. Z. (to appear).Google Scholar
[15]Chi-Te, Tsai, Report on injective modules (Queen's papers in pure and applied mathematics, no. 6; Queen's University, Kingston, Ontario, 1966).Google Scholar
[16]Wilson, John S., “Some properties of groups inherited by normal subgroups of finite index”, Math. Z. 114 (1970), 19–21.CrossRefGoogle Scholar