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On fundamental operations in groups

Published online by Cambridge University Press:  09 April 2009

S. Fajtlowicz
S. Fajtlowicz
Affiliation:
Department of Mathematics, State University of New York at Buffalo, U.S.A.
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By an operation in equationally definable class of group we mean here the element of a free group in this class generated by set {x, y}.

An operation ω(x, y) is called fundamental in a class of groups K if for every group G ω K the operation xy−1 can be expressed in terms of ω.

Higman and Neumann raised in [1] the problem: Is there any binary operation other than xy-1, x-1y, yx-1, y-1x fundamental in the class of all groups?*

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 4 , December 1972 , pp. 445 - 447
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Higman, G. and Neumann, B. H., ‘Groups as groupoids with one law’, Publicationes Mathematicae 2 (1952), 215–221.CrossRefGoogle Scholar
[2]Hulanicki, A. and Świerczkowski, S., ‘On group operations other then xy and yx’ Publicationes Mathematicae 9 (1962), 142–148.CrossRefGoogle Scholar
[3]Padmanabhan, R., ‘A note on inverse binary operation in abelian groups’, Fundamenta Mathematicae LXV (1969), 61–63.CrossRefGoogle Scholar