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
L∗≡L∖{0}, [x,y]≡xyx−1y−1 and cx(y)≡xyx−1
Given g∈K∗ and a∈L∖K
To prove: [g,a]=1
- It is trival that [g,a],[g,a+1] make sense and [g,a]=g(ag−1a−1)∈K so is [g,a+1].
- we have cg(a+1)=g(a+1)g−1=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])
- The assumption [g,a+1]≠[g,a] would lead to a contradiction that a∈K.hence
[g,a+1]≠[g,a]⇔(a+1)g=aga−1(a+1)⇔ga=ag
end of our proof of [g,a]=1.
Let g,h∈K∗,take any ainL∖K.then h+a∈L∖K. We have [g,a+h]=[g,a]=1 hence gh=hg,thus every element of K commutes with L.