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Some remarks on coherent soluble groups

Published online by Cambridge University Press:  17 April 2009

John C. Lennox  and
James Wiegold
John C. Lennox
Affiliation:
Department of Pure Mathematics, University College, Cardiff, U.K.
James Wiegold
Affiliation:
Department of Pure Mathematics, University College, Cardiff, U.K.
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An example of Wehrfritz is pointed out to show that GL(4, Q) is not coherent. This answers a question of Serre. It is shown that finitely generated soluble coherent subgroups of GL(2, Q) need not be polycyclic, in sharp contrast to the fact that all soluble subgroups of GL(n, Z) are polycyclic and so automatically coherent.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 2 , April 1974 , pp. 277 - 279
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Robinson, Derek J.S., Finiteness conditions and generalized soluble groups, Part 1 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 62) Springer-Verlag, Berlin, Heidelberg, New York, 1972.CrossRefGoogle Scholar
[2]Scott, G.P., “Finitely generated 3-manifold groups are finitely presented”, J. London Math. Soc. (2) 6 (1973), 437–440.Google Scholar
[3]Seere, J.-P., Problem section, Proc. Second Internat. Conf. Theory of Groups, Canberra 1973 (Lecture Motes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, to appear).Google Scholar
[4]Wehrfritz, B.A.F., Infinite linear groups: An account of the group-theoretic properties of infinite groups of matrices (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 76. Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar