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 be a finite dimensional division algebra having center , and let be a normal subgroup of finite index. Suppose is not solvable. Then we may assume that 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 *. This property includes the requirement that the diameter of the commuting graph should be , but is, in fact, stronger. Another ingredient is to show that if the commuting graph of has Property , then is open with respect to a nontrivial height one valuation of (assuming without loss of generality, as we may, that is finitely generated). After establishing the openness of (when is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of over its prime subfield to eliminate as a possible quotient of , thereby obtaining a contradiction and proving our main result.

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Additional Information

**Andrei S. Rapinchuk**

Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Email:
asr3x@weyl.math.virginia.edu

**Yoav Segev**

Affiliation:
Department of Mathematics, Ben-Gurion University, Beer-Sheva 84105, Israel

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

DOI:
https://doi.org/10.1090/S0894-0347-02-00393-4

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.

Article copyright:
© Copyright 2002
American Mathematical Society