abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to

division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...

s. It says that given two division rings such that ''xKx''⁻¹ is contained in ''K'' for every ''x'' not equal to 0 in ''D'', either ''K'' is contained in the center of ''D'', or . In other words, if the unit group of ''K'' is a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

of the unit group of ''D'', then either or ''K'' is central .

References

* * Theorems in ring theory Hua Luogeng {{Abstract-algebra-stub