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A note on a centrality property of finitely generated soluble groups

Published online by Cambridge University Press:  24 October 2008

J. C. Lennox
J. C. Lennox
Affiliation:
Department of Pure Mathematics, University College, P.O. Box 78, Cardiff CFI 1XL, Great Britain
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Extract

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.

In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 1 , January 1974 , pp. 23 - 24
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Hall, P.Finiteness conditions in soluble groups. Proc. London Math. Soc. (3), 4 (1954), 419–436.Google Scholar
(2)Kurosh, A. G.Theory of Groups, vol. 2 (Chelsea; New York, 1961).Google Scholar
(3)Lennox, J. C. and Roseblade, J. E.Centrality in finitely generated soluble groups, J. Algebra 16 (1970), 399–438.CrossRefGoogle Scholar
(4)Lnnox, J. C.On a centrality property of finitely generated torsion free soluble groups. J. Algebra 18 (1971), 541–548.CrossRefGoogle Scholar