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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on almost subnormal subgroups of linear groups


Author: B. A. F. Wehrfritz
Journal: Proc. Amer. Math. Soc. 117 (1993), 17-21
MSC: Primary 20E15; Secondary 20G15
DOI: https://doi.org/10.1090/S0002-9939-1993-1119266-4
MathSciNet review: 1119266
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Abstract: Following Hartley we say that a subgroup $ H$ of a group $ G$ is almost subnormal in $ G$ if there is a series of subgroups $ H = {H_0} \leqslant {H_1} \leqslant \cdots \leqslant {H_r} = G$ of $ G$ of finite length such that for each $ i < r$ either $ {H_i}$ is normal in $ {H_{i + 1}}$ or $ {H_i}$ has finite index in $ {H_{i + 1}}$. We extend a result of Hartley's on arithmetic groups (see Theorem $ 2$ of Hartley's Free groups in normal subgroups of unit groups and arithmetic groups, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 173-177) to arbitrary linear groups. Specifically, we prove: let $ G$ be any linear group with connected component of the identity $ {G^0}$ and unipotent radical $ U$. If $ H$ is any soluble-by-finite, almost subnormal subgroup of $ G$ then $ [H \cap {G^0},{G^0}] \leqslant U$.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1119266-4
Article copyright: © Copyright 1993 American Mathematical Society