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Let $KG$ denote the group algebra of the group $G$ over the field $K$ and let$U(KG)$ denote its group of units. Here without the use of a computer we givepresentations for the unit groups of all group algebras $KG$, where the size of$KG$ is less than 1024. As a consequence we find the minimum counterexample tothe Isomorphism Problem for group algebras.

The Unit Group of Small Group Algebras and the Minimum Counterexample to the Isomorphism Problem