Cartan–Brauer–Hua theorem

In abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It says that given two division rings K ⊆ D 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 K = D. In other words, if the unit group of K is a normal subgroup of the unit group of D, then either K = D or K is central (Lam 2001, p. 211).

References

• Herstein, I. N. (1975). Topics in algebra. New York: Wiley. p. 368. ISBN 0-471-01090-1. 
• Lam, Tsit-Yuen (2001). A First Course in Noncommutative Rings (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-95325-0. MR 1838439.