Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-05T02:11:45.440Z Has data issue: false hasContentIssue false
  • English
  • Français

Nilpotent injectors in finite groups

Published online by Cambridge University Press:  17 April 2009

Peter Förster
Peter Förster
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria. 3168, Australia.
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.

Nilpotent injectors exist in all finite groups.

For every Fitting class F of finite groups (see [2]), InjF(G) denotes the set of all H ≤ G such that for each N ⊴ ⊴ G , H ∩ N is an F -maximal subgroup of N (that is, belongs to F and i s maximal among the subgroups of N with this property). Let W and N* denote the Fitting class of all nilpotent and quasi-nilpotent groups, respectively. (For the basic properties of quasi-nilpotent groups, and of the N*-radical F*(G) of a finite group G3 the reader is referred to [5].,X. %13; we shall use these properties without further reference.) Blessenohl and H. Laue have shown in CJ] that for every finite group G, InjN*(G) = {H ≤ G | H ≥ F*(G) N*-maximal in G} is a non-empty conjugacy class of subgroups of G. More recently, Iranzo and Perez-Monasor have verified InjN(G) ≠ Φ for all finite groups G satisfying G = CG(E(G))E(G) (see [6]), and have extended this result to a somewhat larger class M of finite groups C(see [7]). One checks, however, that M does not contain all finite groups; for example, S5 ε M.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 32 , Issue 2 , October 1985 , pp. 293 - 297
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Blessenohl, D., Laue, H., “Fittingklassen endlicher Gruppen, in denen gewisse Hauptfaktoren einfach sind”, J. Algebra 56, (1979), 516–532.CrossRefGoogle Scholar
[2]Fischer, B., Gaschὔtz, W., Hartley, B., “Injektoren endlicher auflsbarer Gruppen”. Math.Z. 102, (1967), 337–339.CrossRefGoogle Scholar
[3]Frster, P., “Projektive Klassen endlicher Gruppen. IIa.” Publ.Sec.Mat.Univ.Aut.Barcelona 29, (1985). (In the press.)Google Scholar
[4]Glauberman, G., “On the automorphism group of a finite group having no non-identity normal subgroups of odd order.” Math.Z. 93, (1966), 154–160.CrossRefGoogle Scholar
[5]Huppert, B., Blackburn, N., “Finite Groups III”, Grundlehren der mathematischen 243, Springer - Verlag, Wissenschaften Berlin - Heidelberg - New York, (1982).Google Scholar
[6]Iranzo, M.J., Pẻrez-Monasor, F., “Existence of N-injectors in a not central normal Fitting class.” Israel J. Math. 48, (1984), 123–128.CrossRefGoogle Scholar
[7]Iranzo, M. J., Pẻrez-Monasor, F., “A class of finite groups having nilpotent injectors.” J. Austral. Math. Soc. (to appear).Google Scholar