## A structure theorem for subgroups of $GL_n$ over complete local Noetherian rings with large residual image

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## Abstract:

Given a complete local Noetherian ring $(A,\mathfrak {m}_A)$ with finite residue field and a subfield $\boldsymbol {k}$ of $A/\mathfrak {m}_A$, we show that every closed subgroup $G$ of $GL_n(A)$ such that $G\mod {\mathfrak {m}_A}\supseteq SL_n(\boldsymbol {k})$ contains a conjugate of $SL_n(W(\boldsymbol {k})_A)$ under some small restrictions on $\boldsymbol {k}$. Here $W(\boldsymbol {k})_A$ is the closed subring of $A$ generated by the Teichmüller lifts of elements of the subfield $\boldsymbol {k}$.## References

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## Additional Information

**Jayanta Manoharmayum**- Affiliation: School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, United Kingdom
- Email: J.Manoharmayum@sheffield.ac.uk
- Received by editor(s): April 5, 2013
- Received by editor(s) in revised form: July 3, 2013
- Published electronically: March 11, 2015
- Communicated by: Pham Huu Tiep
- © Copyright 2015
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**143**(2015), 2743-2758 - MSC (2010): Primary 20E18, 20E34, 11E57
- DOI: https://doi.org/10.1090/S0002-9939-2015-12306-4
- MathSciNet review: 3336600