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Free Variable Axioms for Groups

Published online by Cambridge University Press:  03 November 2016

R. L. Goodstein
R. L. Goodstein*
Affiliation:
University of Leicester
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Extract

The growing importance of the elements of group theory in sixth form teaching has been recognised by a number of recent publications, but a very simple presentation of the theory of groups by means of free variable axioms which was introduced by Lorenzen more than twenty-five years ago, and which is particularly well suited for elementary teaching, does not appear to be as well known as it deserves to be.

Lorenzen defines a group in terms of division x/y, instead of multiplication, a device which obviates the need to postulate the existence of a neutral element and an inverse.

Type
Research Article
Information
The Mathematical Gazette , Volume 52 , Issue 382 , December 1968 , pp. 342 - 347
Copyright
Copyright © Mathematical Association 1968

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References

1. Higman, G. and Neumann, B. H. Groups as groupoids with one law. Publicationes Mathematicae, Debrecen. Vol. 2, 1951-52, pp. 215–221.Google Scholar
2. Lorenzen, P. Ein vereinfachtes Axiomsystem für Gruppen, Journal fur die reine und angewandte Mathematik. Vol. 182 (1940), p. 50.Google Scholar
3. Morgado, J. Another definition of a group by means of a single axiom. Gazeta de Matematica, Lisboa, 1968, pp. 11–13.Google Scholar
4. Sholander, M. Postulates for commutative groups. American Mathematical Monthly. 66, 1959, pp. 93–95.Google Scholar