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 ?
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.
