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Uncountably many Varieties of Groups

Uncountably many Varieties of Groups M. R. VAUGHAN-LEE A variety is said to have the finite basis property if it is determined by a finite set of laws. The study of varieties of groups was begun by B. H. Neumann, who asked in [1] whether the laws of every group are consequences of a finite set of laws of that group or, equivalently, whether every variety of groups has the finite basis property. Many results have been obtained which show that certain classes of varieties do have the finite basis property. R. C. Lyndon [2] showed that the laws of nilpotent varieties are finitely based, and D. E. Cohen [3] proved that metabelian varieties have the finite basis property. In 1963 Sheila Oates and M. B. Powell [4] proved that the laws of a finite group are finitely based. Then in 1969 A. Olshanskii [5] proved that there are 2 ° varieties of groups which are soluble of derived length 5 and of exponent Spq (p, q distinct odd primes). It follows immediately from this that there are varieties of groups whose laws are not finitely based. In this paper, using methods different from A. Olshanskii's, I shall prove the following theorem. x x http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/2.3.280
Publisher site
See Article on Publisher Site

Abstract

M. R. VAUGHAN-LEE A variety is said to have the finite basis property if it is determined by a finite set of laws. The study of varieties of groups was begun by B. H. Neumann, who asked in [1] whether the laws of every group are consequences of a finite set of laws of that group or, equivalently, whether every variety of groups has the finite basis property. Many results have been obtained which show that certain classes of varieties do have the finite basis property. R. C. Lyndon [2] showed that the laws of nilpotent varieties are finitely based, and D. E. Cohen [3] proved that metabelian varieties have the finite basis property. In 1963 Sheila Oates and M. B. Powell [4] proved that the laws of a finite group are finitely based. Then in 1969 A. Olshanskii [5] proved that there are 2 ° varieties of groups which are soluble of derived length 5 and of exponent Spq (p, q distinct odd primes). It follows immediately from this that there are varieties of groups whose laws are not finitely based. In this paper, using methods different from A. Olshanskii's, I shall prove the following theorem. x x

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1970

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