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On Free Groups of the Variety AN 2 ∧ N 2 A

Published online by Cambridge University Press:  20 November 2018

Chander Kanta Gupta
Chander Kanta Gupta*
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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Let R be a commutative ring with unity and let M(R) be the multiplicative group of 4 x 4 triangular matrices (a ij ) over R, where a 11 is a unit element of R and a ii = 1 for i = 2, 3, 4. If V(=AN 2 ∧ N 2 A) denotes the variety of groups which are both abelian-by-class-2 and class-2-by-abelian, then it is routine to verify that M(R) ∊ V. Here we prove the following,

Theorem. Let F(V) denote the free group of finite or countable infinite rank of the variety V. Then for a suitable choice of R, F(V) is embedded in M(R).

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 4 , December 1970 , pp. 443 - 446
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Magnus, Wilhelm, On a theorem of Marshall Hall, Ann. of Math. 40 (1939), 764-768.Google Scholar
2. Neumann, Hanna, Varieties of groups, Springer-Verlag, New York, 1967.Google Scholar