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

Abstract

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let D D be a finite dimensional division algebra having center K K , and let N ⊆ D × N\subseteq D^{\times } be a normal subgroup of finite index. Suppose D × / N D^{\times }/N is not solvable. Then we may assume that H := D × / N 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 1 2 ) (3\frac {1}{2}) . This property includes the requirement that the diameter of the commuting graph should be ≥ 3 \ge 3 , but is, in fact, stronger. Another ingredient is to show that if the commuting graph of D × / N D^{\times }/N has Property ( 3 1 2 ) (3\frac {1}{2}) , then N N is open with respect to a nontrivial height one valuation of D D (assuming without loss of generality, as we may, that K K is finitely generated). After establishing the openness of N N (when D × / N D^{\times }/N is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of K K over its prime subfield to eliminate H H as a possible quotient of D × D^{\times } , thereby obtaining a contradiction and proving our main result.