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A property of finitely generated residually finite groups

Published online by Cambridge University Press:  17 April 2009

P.F. Pickel
P.F. Pickel
Affiliation:
Department of Mathematics, Polytechnic Institute of New York, Brooklyn, New York, USA.
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Abstract

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Let F(G) denote the set of isomorphism classes of finite quotients of the group G. We say that groups G and H have isomorphic finite quotients (IFQ) if F(G) = F(H). In this note, we show that a finitely generated residually finite group G cannot have the same finite quotients as a proper homomorphic image (G is IFQ hopfian). We then obtain some results on groups with the same finite quotients as a relatively free group.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 3 , December 1976 , pp. 347 - 350
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Baumslag, Gilbert, “Groups with the same lower central sequence as a relatively free group. I The groups”, Trans. Amer. Math. Soc. 129 (1967), 308–321.Google Scholar
[2]Baumslag, Gilbert, “Groups with the same lower central sequence as a relatively free group. II. Properties”, Trans. Amer. Math. Soc. 142 (1969), 507–538.CrossRefGoogle Scholar
[3]Neumann, Hanna, Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 37. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar