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A finite basis theorem for product varieties of groups

Published online by Cambridge University Press:  17 April 2009

M.S. Brooks ,
L.G. Kovács  and
M.F. Newman
M.S. Brooks
Affiliation:
Australian National University, Canberra, ACT.
L.G. Kovács
Affiliation:
Australian National University, Canberra, ACT.
M.F. Newman
Affiliation:
Australian National University, Canberra, ACT.
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Abstract

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It is shown that, if U is a subvariety of the join of a nilpotent variety and a metabelian variety and if V is a variety with a finite basis for its laws, then UV also has a finite basis for its laws. The special cases U nilpotent and U metabelian have been established by Higman (1959) and Ivanjuta (1969) respectively. The proof here, which is independent of Ivanjuta's, depends on a rather general sufficient condition for a product variety to have a finite basis for its laws.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 1 , February 1970 , pp. 39 - 44
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Cohen, D.E., “On the laws of a metabelian variety”, J. Algebra 5 (1967), 267–273.CrossRefGoogle Scholar
[2]Higman, Graham, “Some remarks on varieties of groups”, Quart. J. Math. Oxford (2) 10 (1959), 165–178.CrossRefGoogle Scholar
[3]Ivanjuta, I.D., “On bases for the identical relations of certain varieties” (Russian), Mat. Zametki 5 (1969), 137–144.Google Scholar
[4]Neumann, Hanna, Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[5]Pickert, Günter, “Zur Übertragung der Kettensätze”, Math. Ann. 121 (1949), 100–102.CrossRefGoogle Scholar
[6]Ward, M.A., “Basic commutators”, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 343–412.Google Scholar