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A Class of Loops Which are Isomorphic to all Loop Isotopes

Published online by Cambridge University Press:  20 November 2018

Edgar G. Goodaire  and
D. A. Robinson
Edgar G. Goodaire
Affiliation:
Memorial University of Newfoundland, St. John's, Newfoundland
D. A. Robinson
Affiliation:
Georgia Institute of Technology, Atlanta, Georgia
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It is convenient and not without precedent (see [2], [1], and also [6]) to call a loop which is isomorphic to all of its loop isotopes a G-loop. Since all groups are readily seen to be G-loops, the only interest in such loops, from a loop-theoretic standpoint, resides with those which are not associative. Examples and ad hoc constructions of such loops have appeared sporadically in the literature (see, for instance, [1], [2], [4], [6], [8], [9], and [13]).

Any finite loop of order n < 5 is a group and, hence, must also be a G-loop. R. L. Wilson [11, 12, 13] proved that a finite G-loop of prime order is necessarily a group; he also exhibited for each even integer n > 5 a G-loop of order n which is not associative and then raised questions concerning the existence of finite G-loops which are not groups for every possible composite order n > 5.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 662 - 672
Copyright
Copyright © Canadian Mathematical Society 1982

References

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