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On quasinormal subgroups of certain finitely generated groups

Published online by Cambridge University Press:  20 January 2009

John C. Lennox
John C. Lennox
Affiliation:
University CollegeCardiff, Wales
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A subgroup Q of a group G is called quasinormal in G if Q permutes with every subgroup of G. Of course a quasinormal subgroup Q of a group G may be very far from normal. In fact, examples of Iwasawa show (for a convenient reference see [8]) that we may have Q core-free and the normal closure QG of Q in G equal to G so that Q is not even subnormal in G. We note also that the core of Q in G, QG, is of infinite index in QG in this example. If G is finitely generated then any quasinormal subgroup Q is subnormal in G [8] and although Q is not necessarily normal in G we have that |QG:Q| is finite and |QG:Q| is a nilpotent group of finite exponent [5].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 1 , February 1983 , pp. 25 - 28
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

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