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Question: Is it possible to recover a finite group from it's group algebra over $\mathbb{Z}$?

More precisly. By $G_1$ and $G_2$ we denote two finite groups. Let $\mathbb{Z} ( G_1 )$ and $\mathbb{Z} ( G_2 )$ be group algebras. Suppose that $\mathbb{Z} ( G_1 )$ is isomorphic $\mathbb{Z} ( G_2 )$ as algebras. Is $G_1$ isomorphic to $G_2$?

Comment. It is quite obvious that there are not isomorphic finite groups $G_1$ and $G_2$ such that $\mathbb{C} ( G_1 )$ is isomorphic $\mathbb{C} ( G_2 )$. For instance two abelian groups of the same order.

quinque
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  • A related question: http://math.stackexchange.com/questions/368722/what-does-the-group-ring-mathbbzg-of-a-finite-group-know-about-g – H.Durham Mar 23 '15 at 21:13

1 Answers1

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This question has a long history in the mathematical literature. It was probably first suggested by G. Higman in 1941 ( who proved that a Finite Abelian group is determined by its integral group ring). Non-isomorphic finite groups with isomorphic group rings over the integers have been constructed by Martin Hertweck, and the paper was published in Annals of Mathematics around 2000.

Geoff Robinson
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    The paper is: M. Hertweck "A counterexample to the isomorphism problem for integral group rings" Ann. Math. 154 (2001) 1-26. Also Milies and Sehgal's "An Introduction to Group Rings" has a good discussion of the isomorphism problem. – vuur Mar 23 '15 at 21:28