STATEMENT OF THE RESULTS

Ever since Graham Higman's notable thesis [4] the “isomorphism problem for integral group rings” has withstood many attacks:

Given two finite groups G and H, is it true that ℤG ≃ℤH implies G = H? Higman gave a very strong positive answer, in case G is abelian:

(1.1) Every finite subgroup ∪ in the normalized units V(ℤG) - i.e. units of augmentation one - is already a subgroup of G.

Obviously, one can not expect such a strong statement in general, since for any subgroup H of G and a unit u € ℤ G, uHu-1 is a finite subgroup of V(ℤG). Therefore the most one could ask in general is:

(1.2) Is every finite subgroup ∪ of V(ℤG) conjugate in V(ℤG) to a subgroup of G.

But already the dihedral group of order 8,D8, has in V(ℤD8) two conjugacy classes of D8's. It was Berman and Rossa [1] who in 1966 speculated about (1.2), in case G is a finite p-group and ℤ is replaced by ℤp, the ring of p-adic integers, and Ju|∪|=|G|. There are similar considerations in the last chapter of Whitcomb's thesis [6], which he notes as inspired by his advisor, John Thompson.

Our main result in its basic form is a positive answer to this question of Berman and Rossa:

Theorem 1.3. Let G be a finite p-group and H any subgroup of V(ℤp G) with |H|=|G|, then H is conjugate to G by an inner automorphism of ℤpG.