Multiplicative Jordan decomposition in group rings of 3-groups

In this paper, we essentially classify those finite 3-groups G having integral group rings with the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ℤ[G] satisfies MJD. Thus, we are only concerned with the nonabelian case. Here we show that ℤ[G] has the MJD property for the two nonabelian groups of order 3³. Furthermore, we show that there are at most three other specific nonabelian groups, all of order 3⁴, with ℤ[G] having the MJD property. Unfortunately, we are unable to decide which, if any, of these three satisfies the appropriate condition.