There exist two different finite groups with isomorphic integral group rings

Algebra 1 Minute

You need to know: Groups, finite groups, rings, isomorphism for groups and rings.

Background: For any (finite) group G, an integral group ring is the ring whose elements are all possible sums \sum\limits_{g\in G}a_gg, where a_g are integers, addition is defined as \sum_{g\in G}a_gg + \sum_{g\in G}b_gg = \sum_{g\in G}(a_g+b_g)g, and multiplication is given by (\sum_{g\in G}a_gg)\cdot(\sum_{h\in G}b_hh) = \sum_{g\in G}\sum_{h\in G}a_gb_hgh.

The Theorem: On 15th September 1999, Martin Hertweck submitted to Annals of Mathematics a paper in which he proved that there exist two finite non-isomorphic groups G and H such that their integral group rings are isomorphic.

Short context: Obviously, if two groups G and H are isomorphic, then so are their integral group rings. In 1940, Higman asked if the converse is true, that is, whether different finite groups have different integral group rings. The question was answered positively in many special cases. The Theorem, however, implies that in general the answer is negative.

Links: The original paper is here. See also Section 1.5 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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