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A Generalization of a Theorem of Zassenhaus

Published online by Cambridge University Press:  20 November 2018

Steve Ligh
Steve Ligh*
Affiliation:
Texas A. & M University
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A near-ring is a triple (R, +,.) such that (R, +) is a group, (R,.) is a semigroup and. is left distributive over +; i.e. w(x + z) = wx + wz for each w, x, z in R. A near-field is a nearring such that the nonzero elements form a group under multiplication. Zassenhaus [3] showed that if R is a finite near-field, then (R, + ) is abelian. B.H. Neumann [1] extended this result to all near-fields. Recently another proof of this important result was given by Zemmer [4]. The purpose of this note is to give another generalization of the Zassenhaus theorem. In fact, we shall prove the following.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 5 , October 1969 , pp. 677 - 678
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Neumann, B.H., On the commutativity of addition. J. London Math. Soc. 15 (1940) 203–208.Google Scholar
2. Scott, W.R., Group theory. (Prentice Hall, 1964).Google Scholar
3. Zassenhaus, H., Über endlich Fastkörper. Abh. Math. Sem., Univ. Hamburg 11 (1936) 187–220.Google Scholar
4. Zemmer, J. L., The additive group of an infinite near-field is abelian. J. London Math. Soc. 44 (1969) 65–67.Google Scholar