Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T07:41:12.469Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on groups with non-central norm

Published online by Cambridge University Press:  18 May 2009

R. A. Bryce  and
L. J. Rylands
R. A. Bryce
Affiliation:
Department of Mathematics, Faculty of Science, Australian National University, Canberra Act 0200
L. J. Rylands
Affiliation:
Department of Mathematics, University of Western Sydney, Nepean, P.O. Box 10, Kingswood NSW 2747
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.

The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G. Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 1 , January 1994 , pp. 37 - 43
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Baer, R., Der Kern 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., Cossey, John and Ormerod, E. A., A note on p-groups with power automorphisms. Glasgow Math. J. 34 (1992), 327–332.CrossRefGoogle Scholar
4.Cannon, J. J., An introduction to the Group Theory Language Cayley. Computational Group Theory (Durham 1982) pp. 145–183, (Academic Press, London–New York, 1984).Google Scholar
5.Dedekind, R., Über Gruppen, deren sämmtliche Theiler Normaltheiler sind. Math. Ann. 48 (1897), 548–561.CrossRefGoogle Scholar
6.Hall, M., The Theory of Groups.(Macmillan, New York, 1959).Google Scholar
7.Hughes, D., Research Problem No. 3, Bull. Amer. Math. Soc. 63 (1957), 209.Google Scholar
8.Levi, F. W., Groups in which the commutator operation satisfies certain algebraic conditions. J. Indian Math. Soc. 6 (1942), 87–97.Google Scholar
9.Meixner, Thomas, Power automorphisms of finite p-groups, Israel J. Math. 38 (1981), 345–360.CrossRefGoogle Scholar
10.Schenkman, E., On the norm of a group. Illinois J. Math. 4 (1960), 150–152.CrossRefGoogle Scholar
11.Strauss, E. G. and Szekeres, G., On a problem of D. R. Hughes. Proc. Amer. Math. Soc. 9 (1958), 157–158.CrossRefGoogle Scholar