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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.
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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

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