The ‘isomorphism problem for integral group rings’
(IP) is the question whether
for two finite groups G and H, the existence of
an isomorphism of the integral group
rings ℤG and ℤH implies that G and
H
are isomorphic. Though (IP) is not true
in general [4], it is still an interesting question
for which classes of finite groups
(IP) has a positive solution. In this note, we want to show that (IP) holds
for finite
groups of Lie type associated to Chevalley groups of universal type.
Note that if U(ℤG) denotes the units of ℤG,
and
V(ℤG) stands for the units with augmentation 1,
then
U(ℤG)≅V(ℤG)×U(ℤ),
and we can always assume that H[les ]V(ℤG).