Multiplicative subgroups of finite index in a division ring

Abstract:

If $G$ is a subgroup of finite index $n$ in the multiplicative group of a division ring $F$ then $G - G = F$ or $|F| < {(n - 1)^4} + 4n$. For infinite $F$ this is derived from the Hales-Jewett theorem. If $|F| > {(n - 1)^2}$ and $- 1$ is a sum of elements of $G$ then every element of $F$ has this property; the bound ${(n - 1)^2}$ is optimal for infinitely many $n$.