## Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

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- by Andrei S. Rapinchuk, Yoav Segev and Gary M. Seitz
- J. Amer. Math. Soc.
**15**(2002), 929-978 - DOI: https://doi.org/10.1090/S0894-0347-02-00393-4
- Published electronically: June 21, 2002
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## Abstract:

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let $D$ be a finite dimensional division algebra having center $K$, and let $N\subseteq D^{\times }$ be a normal subgroup of finite index. Suppose $D^{\times }/N$ is not solvable. Then we may assume that $H:=D^{\times }/N$ is a*minimal nonsolvable group*(MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote

*Property $(3\frac {1}{2})$*. This property includes the requirement that the diameter of the commuting graph should be $\ge 3$, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of $D^{\times }/N$ has Property $(3\frac {1}{2})$, then $N$ is open with respect to a nontrivial height one valuation of $D$ (assuming without loss of generality, as we may, that $K$ is finitely generated). After establishing the openness of $N$ (when $D^{\times }/N$ is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of $K$ over its prime subfield to eliminate $H$ as a possible quotient of $D^{\times }$, thereby obtaining a contradiction and proving our main result.

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## Bibliographic Information

**Andrei S. Rapinchuk**- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 206801
- Email: asr3x@weyl.math.virginia.edu
**Yoav Segev**- Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel
- MR Author ID: 225088
- Email: yoavs@math.bgu.ac.il
**Gary M. Seitz**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1226
- Email: seitz@math.uoregon.edu
- Received by editor(s): February 28, 2001
- Received by editor(s) in revised form: January 24, 2002
- Published electronically: June 21, 2002
- Additional Notes: The first author was partially supported by grants from the NSF and by BSF grant no.Â 97-00042

The second author was partially supported by BSF grant no.Â 97-00042. Portions of this work were written while the author visited the Forschungsinstitut fĂŒr Mathematik ETH, Zurich, in the summer of 2000, and the author gratefully acknowledges the hospitality and support.

The third author was partially supported by grants from the NSF and by BSF grant no.Â 97-00042. - © Copyright 2002 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**15**(2002), 929-978 - MSC (1991): Primary 16K20, 16U60; Secondary 20G15, 05C25
- DOI: https://doi.org/10.1090/S0894-0347-02-00393-4
- MathSciNet review: 1915823