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*, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 173-177. MR**1003352 (90g:20006)****[2]**B. A. F. Wehrfritz,*Infinite linear groups*, Springer-Verlag, Berlin, 1973. MR**0335656 (49:436)****[3]**-,*On the Lie-Kolchin-Mal'cev theorem*, J. Austral. Math. Soc. Ser. A**26**(1978), 270-276. MR**515743 (80c:20070)****[4]**-,*Lectures around complete local rings*, Queen Mary College Math. Notes, London, 1979. MR**550883 (80k:13014)****[5]**-,*Wielandt's subnormality criterion and linear groups*, J. Algebra**67**(1980), 491-503. MR**602076 (82g:20055)**

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

Article copyright:
© Copyright 1993
American Mathematical Society