Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-06T09:45:49.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Local endomorphism near-rings

Published online by Cambridge University Press:  20 January 2009

Carter G. Lyons  and
Gary L. Peterson
Carter G. Lyons
Affiliation:
Department of Mathematics and Computer ScienceJames Madison UniversityHarrisonburg, VA., 22807, U.S.A.
Gary L. Peterson
Affiliation:
Department of Mathematics and Computer ScienceJames Madison UniversityHarrisonburg, VA., 22807, U.S.A.
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 purpose of this paper is to study the consequences of an endomorphism near-ring of a finite group being a local near-ring and the existence of such near-rings. As we shall see in Section 2, an endomorphism near-ring of a finite group being local gives us some information about both the structure of the group (Theorem 2.2) and the automorphisms of the group lying in the near-ring (Theorem 2.3). Existence of local endomorphism near-rings of finite groups is considered in Section 3 where we obtain as our main result that any p-group of automorphisms of a p-group containing the inner automorphisms always generates a local endomorphism near-ring. In particular, we get as a corollary that the endomorphism near-ring of a finite group G generated by the inner automorphisms of G is local if and only if G is a p-group. The third section concludes with a discussion of endomorphism near-rings of dihedral 2-groups and generalized quaternion groups.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 3 , October 1988 , pp. 409 - 414
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

REFERENCES

1.Gorenstein, D., Finite Groups (Harper and Row, New York, 1968).Google Scholar
2.Hall, M., The Theory of Groups (Macmillan, New York, 1959).Google Scholar
3.Lyons, C. G. and Malone, J. J., Finite dihedral groups and d.g. near rings II, Compositio Math. 26 (1973), 249–259.Google Scholar
4.Lyons, C. G. and Meldrum, J. D. P., Characterizing series for faithful d.g. near-rings, Proc. Amer. Math. Soc. 72 (1978), 221–227.Google Scholar
5.Malone, J. J., Generalised quaternion groups and distributively generated near-rings, Proc. Edinburgh Math. Soc. 18 (1973), 235–238.CrossRefGoogle Scholar
6.Maxson, C. J., On local near-rings, Math. Z. 106 (1968), 197–205.CrossRefGoogle Scholar
7.Meldrum, J. D. P., Near-Rings and Their Links with Groups (Pitman, London, 1985).Google Scholar
8.Schmid, P., Über die stabilitätsgruppen der untergruppenreihen einer endlichen gruppe, Math. Z. 123 (1971), 318–324.CrossRefGoogle Scholar
9.Zassenhaus, H. J., The Theory of Groups, second edition (Chelsea, New York, 1958).Google Scholar