---
tag: KC22CWGGAE
kind: statement
role: theorem
name: Cartan–Brauer–Hua theorem
msc: [16K40, 16U70]
uses: [0WZXK05ME7, 0859D58WPD, MVT64B3B6H]
about: [0EA5BTA0CX]
sources:
  - url: https://en.wikipedia.org/wiki/Cartan%E2%80%93Brauer%E2%80%93Hua_theorem
same_as:
  wikidata: Q5047046
  wikipedia: Cartan–Brauer–Hua theorem
---

## Statement

Let $D$ be a [[division ring|0WZXK05ME7]], and let $K \subseteq D$ be a division [[subring|0859D58WPD]] of $D$.
Assume that $x K x^{-1} \subseteq K$ for every nonzero $x \in D$.
Then $K = D$ or $K$ is contained in the [[center|MVT64B3B6H]] $Z(D) = \set{z \in D : zd = dz \text{ for all } d \in D}$.

## Remarks

A short proof works with two conjugates.
Take $a \in K$ and $b \in D \setminus K$; then $b \ne 0$ and $1 + b \ne 0$, so $k_1 = b a b^{-1}$ and $k_2 = (1+b) a (1+b)^{-1}$ lie in $K$.
Expanding $(1+b)a = k_2(1+b)$ and using $ba = k_1 b$ gives $(k_1 - k_2) b = k_2 - a$, so $k_1 = k_2$ (otherwise $b \in K$), hence $a = k_2 = k_1$ and $ab = ba$.
So every element of $K$ commutes with $D \setminus K$; if $K \ne D$, every element of $K$ is a difference of two elements of $D \setminus K$, so it also commutes with $K$, and $K \subseteq Z(D)$.
Example: in the real quaternions $\mathbb{H}$, the subfield $\C = \R + \R i$ is neither $\mathbb{H}$ nor central, and indeed $(1+j)\, i \,(1+j)^{-1} = -k \notin \C$.

## Explanations

### L1 human

Conjugating by $x$ means replacing $y$ by $x y x^{-1}$.
The theorem says that a division subring which is stable under all conjugations is either everything or sits inside the center.
So a skew field has no "normal" sub-skew-fields other than the trivial ones.

### L2 human

The theorem is a division-ring analogue of the fact that a normal subgroup is a union of conjugacy classes.
It is often applied with $K$ the subring generated by a conjugation-invariant set, such as the center together with some commutators.
Its proof is elementary, in contrast with deeper results such as [[Wedderburn's little theorem|0RRNVX1VKA]], and it has a companion in [[Hua's identity|AWS7XTZ49C]], another elementary tool for division rings.