A Continuous Proof of Zassenhaus’s Solubility Theorem

Abstract A class $$\mathcal {S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> of soluble groups is D-bounded when there exists a uniform upper bound for the lengths $$d(\varGamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the derived series for $$\varGamma \in \mathcal {S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> . A theorem of Zassenhaus (Abh. Math. Semin. Hansisch. Univ. 12 , 289–312, 1938) states that for each n the class of soluble subgroups of $$GL(n,\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is D -bounded. Although Zassenhaus’s theorem is fundamental to the study infinite discrete linear groups the proof given here is located within the theory of continuous groups and the only discrete groups which appear are finite.