Wedderburn's little theorem

Wedderburn’s little theorem: Every finite division ring is commutative. Equivalently, every finite division ring is a field .

A standard approach studies the finite group $D^\times$ of units, i.e. the group of units , and compares its conjugacy structure with the center of the ring using the class equation .

Proof sketch: Let $D$ be finite with center $Z$, so $Z$ is a finite field. The conjugation action of $D^\times$ on itself partitions $D^\times$ into conjugacy classes whose sizes are indices of centralizers, and each centralizer is a $Z$-subalgebra. Counting conjugacy classes and exploiting that $|Z^\times|$ divides centralizer sizes forces every element to commute with $Z$ in a way that ultimately yields $[D:Z]=1$, hence $D=Z$ is commutative.