A Counterexample to the Isomorphism Problem for Integral Group Rings

Let X be a finite group, and denote its integral group ring by ZX. A group basis of ZX is a subgroup Y of the group of units of ZX of augmentation 1 such that ZX = ZY and IXI = YI. An example of a finite group X is given such that ZX has a group basis which is not isomorphic to X. A main ingredient is the existence of a subgroup G of X which possesses a non-inner automorphism which becomes inner in the integral group ring ZG. The question whether a finite group X is determined by its integral group ring ZX is known as the 'isomorphism problem for integral group rings'. It was