Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-15T03:49:21.883Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on p-groups with power automorphisms

Published online by Cambridge University Press:  18 May 2009

R. A. Bryce ,
John Cossey  and
E. A. Ormerod
R. A. Bryce
Affiliation:
Department of Mathematics, School of Mathematical Sciences, Australian National University, P. O. Box 4, Canberra City, Act 2601, Australia
John Cossey
Affiliation:
Department of Mathematics, School of Mathematical Sciences, Australian National University, P. O. Box 4, Canberra City, Act 2601, Australia
E. A. Ormerod
Affiliation:
Department of Mathematics, School of Mathematical Sciences, Australian National University, P. O. Box 4, Canberra City, Act 2601, Australia
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.

Let G be a group. The norm, or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G. In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω(G). The Wielandt series of subgroups ω1(G) is defined by: ω1(G) = ω(G) and for i ≥ 1, ωi+1(G)/ ω(G) = ω(G/ωi, (G)). The subgroups of the upper central series we denote by ζi(G).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 3 , September 1992 , pp. 327 - 332
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Der Kern, R. Baer, eine charakteristische Untergruppe, Compositio Math. 1 (1935), 254–283.Google Scholar
2.Baer, R., Gruppen mit Hamiltonschen Kern, Compositio Math. 2 (1935), 241–246.Google Scholar
3.Bryce, R. A. and Cossey, John, The Wielandt subgroup of a finite soluble group, J. London Math. Soc. (2) 40 (1989), 244–256.CrossRefGoogle Scholar
4.Dedekind, R., Über Gruppen deren samtliche Teiler Normalteiler sind, Math. Ann. 48 (1897), 548–561.CrossRefGoogle Scholar
5.Gupta, N. D. and Newman, M. F., On metabelian groups, J. Austral. Math. Soc. 6 (1966), 362–368.CrossRefGoogle Scholar
6.Liebeck, Hans, Concerning nilpotent wreath products, Proc. Cambridge Philos. Soc. 6 (1962), 443–451.CrossRefGoogle Scholar
7.Meier-Wunderli, H., Metabelsche Gruppen, Comment. Math. Helv. 25 (1951), 1–10.CrossRefGoogle Scholar
8.Meixner, Thomas, Power automorphisms of finite p-groups, Israel J. Math 38 (1981), 345–360.CrossRefGoogle Scholar
9.Schenkman, E., On the norm of a group, Illinois J. Math. 4 (1960), 150–152.CrossRefGoogle Scholar
10.Wielandt, H., Über den Normalisator der subnormalen Untergruppen. Math. Z. 69 (1958), 463–465.CrossRefGoogle Scholar