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A non-cyclic one-relator group all of whose finite quotients are cyclic

To Bernhard Hermann Neumann on his 60th birthday

Published online by Cambridge University Press:  09 April 2009

Gilbert Baumslag
Gilbert Baumslag
Affiliation:
Institute for Advanced Study and Rice University
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Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 3-4 , November 1969 , pp. 497 - 498
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory (Interscience Publishers, 1966).Google Scholar
[2]Baumslag, G. and Solitar, D., ‘Some two-generator one-relator nonhopfian groups’, Bull. Amer. Math. Soc. 68 (1962), 199–201.CrossRefGoogle Scholar
[3]Higman, G., ‘A finitely generated infinite simple group’, J. London Math. Soc. 26 (1951), 61–64.CrossRefGoogle Scholar