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A note on the normalizer condition

Published online by Cambridge University Press:  24 October 2008

B. Hartley
B. Hartley
Affiliation:
University of Warwick
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Extract

The first example of a non-trivial group satisfying the normalizer condition but having trivial centre was obtained by Heineken and Mohamed (1). In fact, the group they constructed satisfies the stronger condition that all its proper subgroups are subnormal and nilpotent. In this note we use a construction suggested by their approach to obtain such a group G as a subgroup of the restricted wreath product Cp ≀ Cp∞. The fact that G is given as a subgroup of a well known group makes it rather easier to investigate its properties, and in particular to see that its centre Z(G) is trivial.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 1 , July 1973 , pp. 11 - 15
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Heineken, H. and Mohamed, I. J.A group with trivial centre satisfying the normalizer condition. J. Algebra 10 (1968), 368–376.CrossRefGoogle Scholar
(2)Heineken, H. and Mohamed, I. J.Groups with normalizer condition. Math. Ann. 198 (1972), 179–188.CrossRefGoogle Scholar
(3)Meldrum, J. D. P.On the Heineken–Mohamed groups. J. Algebra, to appear.Google Scholar