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In 1987 Roggenkamp and Scott published a solution of the integral isomorphism problem for $p$-groups, i.e. if $G,H$ are $p$-groups and $\mathbb{Z}[G] \cong \mathbb{Z}[H]$ as rings then $G \cong H$.

However, in practice I guess it is at least as hard to show that two group rings aren't isomorphic than to show that the groups itself aren't isomorphic. Therefore I wonder if this theorem (or one of its variants or generalizations) have found applications in group theory. Any idea ?

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The Annals paper by Roggenkamp and Scott was certainly a landmark in the ongoing study of the isomorphism problem for integral group rings of finite groups, which apparently goes back to the thesis work of Graham Higman and later related work by Richard Brauer. Zassenhaus refined and extended the underlying problem of whether two finite groups with isomorphic group rings over $\mathbb{Z}$ must necessarily be isomorphic.

I'm not at all a specialist in this line of work, which has spawned numerous papers and at least one book, including positive and negative answers to versions of the original problem. But as far as I know the question itself is mainly theoretical (though quite natural), not likely to have direct concrete applications one way or the other. Rather, the "applications" would involve related areas of integral representation theory and possibly algebraic topology where integral group rings come up naturally.

Eventually in a 2001 Annals paper, Martin Hertweck arrived at a negative answer to the initial problem: see the extensive review by Donald Passman in Mathematical Reviews. But questions of this type continue to be explored.

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Thanks for your answer. Do you know the title of the book on the isomorphism problem you mentioned ? – Todd Leason May 25 '12 at 23:00
My recollection (not precise) is that S.K. Sehgal wrote an older book, which is probably outdated in many directions, but also co-authored a more recent textbook which includes much of the work on group rings and related matters. – Jim Humphreys May 26 '12 at 0:07
Roughly speaking (for one possible direction of application) It is sometimes possible to show the existence of central units in integral or modular group rings which, if they could be shown to be genuine group elements, would give a proof of some interesting general group-theoretic conjectures. – Geoff Robinson May 26 '12 at 5:36
@Geoff: Thanks for the wider perspective on this area of research. I was commenting more narrowly just on the positive solution of the isomorphism problem for group rings given for certain groups by Roggenkamp and Scott (though their work goes farther than this problem). – Jim Humphreys May 26 '12 at 18:40

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