Multiplicative Jordan decomposition in group rings and p-groups with all noncyclic subgroups normal

Let p be a prime and let G be a finite p-group. We show that if the integral group ring Z[G] satisfies the multiplicative Jordan decomposition property, then every noncyclic subgroup of G is normal. This is used to simplify the work of Hales, Passi and Wilson on the classification of integral group rings of finite 2-groups with the multiplicative Jordan decomposition property.