The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

中文翻译：

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二面体和四元数群的群代数比较

比较了广义四元数群和 2 次幂的二面体群的群代数。他们在特征为 2 的有限域上的群代数已知是非同构的，并且给出了一些可能具有独立意义的新证明。然而，这两个群代数非常相似，并且显示出具有许多共同的环论性质。最后，考虑半简单情况（其中场的特征大于 2）并确定同构问题的最小非对易反例。