Separable deformations of the generalized quaternion group algebras

Abstract

The group algebras $k{Q}_{2^n}$ of the generalized quaternion groups $Q_{2^n}$ over fields $k$ which contain ${\mathbb{F}}_{2^{n-2}}$, are deformed to separable $k((t))$-algebras $[k{Q}_{2^n}{]}_t$. The dimensions of the simple components of $\overline{k((t))}{\otimes }_{k((t))}[k{Q}_{2^n}{]}_t$ over the algebraic closure $\overline{k((t))}$, and those of $\mathbb{C}{Q}_{2^n}$ over $\mathbb{C}$ are the same, yielding strong solutions of the Donald-Flanigan conjecture for the generalized quaternion groups.