Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T08:43:15.454Z Has data issue: false hasContentIssue false

Locally soluble groups with min-n.

Published online by Cambridge University Press:  09 April 2009

H. Heineken
Affiliation:
Mathematisches Institut, Universität Erlangen-Nürnberg, 852 Erlangen, Bismarckstr. 1 1/2., Fed. Rep. of Germany
J. S. Wilson
Affiliation:
Christ's College, Cambridge Great Britain
Rights & Permissions [Opens in a new window]

Extract

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.

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Baer, R., ‘Irreducible groups of automorphisms of abelian groups’, Pacific J. Math. 14 (1964), 385406.CrossRefGoogle Scholar
[2]McLain, D. H., ‘Finiteness conditions in locally soluble groups’, J. London Math. Soc. 34 (1959), 101107.CrossRefGoogle Scholar
[3]Neumann, B. H., ‘Twisted wreath products of groups’, Arch. Math. 14 (1963), 16.CrossRefGoogle Scholar
[4]McLain, D. H., ‘On locally nilpotent groups’, Proc. Cambridge Philos. Soc. 52 (1956), 511.CrossRefGoogle Scholar
[5]Hall, P., ‘Finiteness conditions for soluble groups’, Proc. London Math. Soc. 4 (1954), 419436.Google Scholar
[6]Hall, P., ‘Wreath powers and characteristically simple groups’, Proc. Cambridge Philos. Soc. 58 (1962), 170184.CrossRefGoogle Scholar