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The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been done towards solving this and related problems. See for example Chapter 9 in Milies, Sehgal, "An Introduction to Group Rings". Now it seems that the first (?!) counterexample has been found by Martin Hertweck in his 2001 paper "A Counterexample to the Isomorphism Problem for Integral Group Rings". He has constructed two counterexamples, the one group has order $2^{25} \cdot 97^2$ and the other group has $2^{21} \cdot 97^{28}$. Do we really have to consider such huge groups? The thesis by Geoffrey Janssens discusses Hertweck's construction in detail, and claims that this is the only known counterexample. Is this still correct?

Question. What is known about the minimal counterexamples of the isomorphism problem for integral group rings?

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I think, it's correct that the smallest known counterexample is still the Hertweck's pair of non-isomorphic groups of order $2^{21} \cdot 97^{28}$.

Please note that the group of order $2^{25} \cdot 97^2$ is used in the construction, but there is no known pair of groups of that order that provide a counterexample.

Of course, it may happen that a smaller counterexample exists. Remember, for example, the situation with the 2nd Zassenhaus conjecture, where by know we know a variety of counterexamples and the minimal one is 30 times smaller than the first discovered. To give a more detailed account, known counterexamples to the 2nd Zassenhaus conjecture are:

  • of order 2880 – Roggenkamp & Scott, 1988

  • of orders 2880 and 6720 – Klingler, 1991

  • of order 1140 (metabelian) – Hertweck, 2002

  • Hertweck, 2003 – of orders: 180 (metabelian), 360 (supersolvable), 72600 (Frobenius)

  • of order 96 (three groups) – Blanchard, 2001 (who also verified using GAP 3.4.4 that these are counterexamples of minimal possible order)

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  • $\begingroup$ Thank you. Does one know that the isomorphism problem holds for groups of order $\leq n$ for some specific large values of $n$? $\endgroup$ Jan 9 '14 at 13:35
  • $\begingroup$ @MartinBrandenburg: to my best knowledge, no. There were results formulated for classes of groups, but not for groups of consecutive orders smaller than some bound. The Modular Isomorphism Problem stands a bit better, e.g. for $p=2$ it is checked that it holds for all 2-groups of order up to $2^8$ (theoretical proof for up to $2^6$, the rest using computers). $\endgroup$ Jan 9 '14 at 15:51

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