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 of a group is almost subnormal in if there is a series of subgroups of of finite length such that for each either is normal in or has finite index in . We extend a result of Hartley's on arithmetic groups (see Theorem 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 be any linear group with connected component of the identity and unipotent radical . If is any soluble-by-finite, almost subnormal subgroup of then .

**[1]**B. Hartley,*Free groups in normal subgroups of unit groups and arithmetic groups*, Representation theory, group rings, and coding theory, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 173–177. MR**1003352**, https://doi.org/10.1090/conm/093/1003352**[2]**B. A. F. Wehrfritz,*Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76. MR**0335656****[3]**B. A. F. Wehrfritz,*On the Lie-Kolchin-Mal′cev theorem*, J. Austral. Math. Soc. Ser. A**26**(1978), no. 3, 270–276. MR**515743****[4]**B. A. F. Wehrfritz,*Lectures around complete local rings*, Queen Mary College, Department of Pure Mathematics, London, 1979. Queen Mary College Mathematics Notes. MR**550883****[5]**B. A. F. Wehrfritz,*Wielandt’s subnormality criterion and linear groups*, J. Algebra**67**(1980), no. 2, 491–503. MR**602076**, https://doi.org/10.1016/0021-8693(80)90173-8

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1119266-4

Article copyright:
© Copyright 1993
American Mathematical Society