   ISSN 1088-6834(online) ISSN 0894-0347(print)

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

Authors: Andrei S. Rapinchuk, Yoav Segev and Gary M. Seitz
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
Published electronically: June 21, 2002
MathSciNet review: 1915823
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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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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

Keywords: Division algebra, multiplicative group, finite homomorphic images, valuations
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. 97-00042
The second author was partially supported by BSF grant no. 97-00042. Portions of this work were written while the author visited the Forschungsinstitut für 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. 97-00042.