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On multiplicative systems defined by generators and relations

II. Monogenic loops

Published online by Cambridge University Press:  24 October 2008

Trevor Evans
Trevor Evans
Affiliation:
The Institute for Advanced StudyPrinceton, New Jersey*
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Extract

The techniques developed in (9) are used here to study the properties of multiplicative systems generated by one element (monogenie systems). The results are of two kinds. First, we obtain fairly complete information about the automorphisms and endo-morphisms of free and finitely related loops. The automorphism group of the free monogenie loop is the infinite cyclic group, each automorphism being obtained by mapping the generator on one of its repeated inverses. A monogenie loop with a finite, non-empty set of relations has only a finite number of endomorphisms. These are obtained by mapping the generator on some of the components, or their repeated inverses, occurring in the relations. We use the same methods to solve the isomorphism problem for monogenie loops, i.e. we give a method for determining whether two finitely related monogenie loops are isomorphic. The decision method consists essentially of constructing all homomorphisms between two given finitely related monogenie loops.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 4 , October 1953 , pp. 579 - 589
Copyright
Copyright © Cambridge Philosophical Society 1953

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