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RESIDUALLY FINITE GROUPS WITH ALL SUBGROUPS SUBNORMAL

Published online by Cambridge University Press:  01 November 1999

HOWARD SMITH
HOWARD SMITH
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA
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Abstract

The following result is established.

THEOREM. Let G be a periodic, residually finite group with all subgroups sub-normal. Then G is nilpotent.

The well-known groups of Heineken and Mohamed [1] show that the hypothesis of residual finiteness cannot be omitted here, while examples in [5] show that a residually finite group with all subgroups subnormal need not be nilpotent. The proof of the Theorem will use the results of Möhres that a group with all subgroups subnormal is soluble [3] and that a periodic hypercentral group with all subgroups subnormal is nilpotent [4]. Borrowing an idea from [2], the plan is to construct certain subgroups H and K that intersect trivially, and to show that the subnormality of both leads to a contradiction.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 31 , Issue 6 , November 1999 , pp. 679 - 680
Copyright
© The London Mathematical Society 1999

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