# Separable deformations of the generalized quaternion group algebras

## TL;DR

This paper constructs separable deformations of group algebras of generalized quaternion groups over certain fields, matching their simple component dimensions with those over complex numbers, thus solving the Donald-Flanigan conjecture for these groups.

## Contribution

It introduces explicit separable deformations of generalized quaternion group algebras over fields containing specific finite fields, confirming the conjecture for these groups.

## Key findings

- Deformed algebras are separable over $k((t))$.
- Dimensions of simple components match those over $\,\mathbb{C}$.
- Provides strong solutions to the Donald-Flanigan conjecture for generalized quaternion groups.

## Abstract

The group algebras $kQ_{2^n}$ 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 $[kQ_{2^n}]_t$. The dimensions of the simple components of $\overline{k((t))}\otimes_{k((t))}[kQ_{2^n}]_t$ 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.