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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite dimensional division algebra having center <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N subset-of-or-equal-to upper D Superscript times"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>×</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">N\subseteq D^{\times }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a normal subgroup of finite index. Suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript times Baseline slash upper N"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>×</mml:mo> </mml:mrow> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">D^{\times }/N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not solvable. Then we may assume that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H colon equals upper D Superscript times slash upper N"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>:=</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>×</mml:mo> </mml:mrow> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">H:=D^{\times }/N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <italic>minimal nonsolvable group</italic> (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 <italic>Property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 3 and one half right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(3\frac {1}{2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula></italic>. This property includes the requirement that the diameter of the commuting graph should be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript times Baseline slash upper N"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>×</mml:mo> </mml:mrow> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">D^{\times }/N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has Property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 3 and one half right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(3\frac {1}{2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is open with respect to a nontrivial height one valuation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (assuming without loss of generality, as we may, that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finitely generated). After establishing the openness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript times Baseline slash upper N"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>×</mml:mo> </mml:mrow> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">D^{\times }/N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over its prime subfield to eliminate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a possible quotient of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript times"> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>×</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">D^{\times }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, thereby obtaining a contradiction and proving our main result.