Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-29T19:50:17.124Z Has data issue: false hasContentIssue false
  • English
  • Français

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*
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Albert, A. A.Quasigroups. I. Trans. Amer. math. Soc. 54 (1943), 507–19.CrossRefGoogle Scholar
(2)Albert, A. A.Quasigroups. II. Trans. Amer. math. Soc. 55 (1944), 401–19.Google Scholar
(3)Baer, R.Nets and groups. I. Trans. Amer. math. Soc. 46 (1939), 110–41.CrossRefGoogle Scholar
(4)Baer, R.The homomorphism theorems for loops. Amer. J. Math. 67 (1945), 450–60.CrossRefGoogle Scholar
(5)Bates, Grace E.Free loops and nets and their generalizations. Amer. J. Math. 69 (1947), 499–550.CrossRefGoogle Scholar
(6)Bruck, R. H.Simple quasigroups. Bull. Amer. math. Soc. 50 (1944), 769–81.CrossRefGoogle Scholar
(7)Bruck, R. H.Some results in the theory of quasigroups. Trane. Amer. math. Soc. 55 (1944), 19–52.CrossRefGoogle Scholar
(8)Evans, T.The word problem for abstract algebras. J. Lond. math. Soc. 26 (1951), 64–71.Google Scholar
(9)Evans, T.On multiplicative systems defined by generators and relations. I. Proc. Camb. phil. Soc. 47 (1951), 637–49.CrossRefGoogle Scholar
(10)Evans, T.Embedding theorems for multiplicative systems and projective geometries. Proc. Amer. math. Soc. 3 (1952), 614–20.CrossRefGoogle Scholar
(11)Higman, G.A finitely related group with an isomorphic proper factor group. J. Lond. math. Soc. 26 (1951), 59–61.CrossRefGoogle Scholar
(12)Higman, G., Neumann, B. H. and Neumann, H.Embedding theorems for groups. J. Lond. math. Soc. 24 (1949), 247–54.CrossRefGoogle Scholar
(13)Neumann, B. H.Some remarks on infinite groups. J. Lond. math. Soc. 12 (1937), 120–27.CrossRefGoogle Scholar
(14)Neumann, B. H.A two-generator group isomorphic to a proper factor group. J. Lond. math. Soc. 25 (1950), 247–8.CrossRefGoogle Scholar
(15)Neumann, B. H. and Evans, T.Varieties of loops and groupoids. J. Lond. math. Soc. (to appear).Google Scholar