Multiplicative Jordan Decomposition in Group Rings of 3-Groups

Abstract

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.

Multiplicative Jordan Decomposition in Group Rings of 2, 3-Groups

“…Specifically, these are the two nonabelian groups of order 8, five groups of order 16, four groups of order 32, and only the Hamiltonian groups of larger order. On the other hand, in paper [6] we were able to build on the work of [5], using variants of many of the same arguments, to essentially determine all nonabelian 3-groups satisfying MJD. These are the two nonabelian groups of order 3 3 and at most three groups of order 3 4 .…”