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Cartan-Brauer-Hua Theorem

Cartan-Brauer-Hua For a proper subset divison ring K of division ring L, if the unit group of K is a normal subgroup of the unit group of L, K is central.

I will show two similar proof of the Cartan–Brauer–Hua theorem.

First Proof

Definition
LL{0}, [x,y]xyx1y1 and cx(y)xyx1

Given gK and aLK
To prove: [g,a]=1

  1. It is trival that [g,a],[g,a+1] make sense and [g,a]=g(ag1a1)K so is [g,a+1].
  2. we have cg(a+1)=g(a+1)g1=cg(a)+1 hence [g,a+1](a+1)=[g,a]a+1 or equivalently [g,a+1][g,a])a=(1[g,a+1])
  3. The assumption [g,a+1][g,a] would lead to a contradiction that aK.hence
    [g,a+1][g,a](a+1)g=aga1(a+1)ga=ag
    end of our proof of [g,a]=1.

Let g,hK,take any ainLK.then h+aLK. We have [g,a+h]=[g,a]=1 hence gh=hg,thus every element of K commutes with L.

Second Proof by Hua Luogeng

Proof