On integral decomposition of unipotent elements in integral group rings

Abstract Jespers and Sun conjectured in [27] that if a finite group G has the property ND, i.e. for every nilpotent element n in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in \mathbb{Z}G$ , then at most one of the simple components of the group algebra $\mathbb{Q} G$ has reduced degree bigger than 1. With the exception of one very special series of groups we are able to answer their conjecture, showing that it is true—up to exactly one exception. To do so, we first classify groups with the so-called SN property which was introduced by Liu and Passman in their investigation of the Multiplicative Jordan Decomposition for integral group rings. The conjecture of Jespers and Sun can also be formulated in terms of a group q ( G ) made from the group generated by the unipotent units, which is trivial if and only if the ND property holds for the group ring. We answer two more open questions about q ( G ) and notice that this notion allows to interpret the studied properties in the general context of linear semisimple algebraic groups. Here we show that q ( G ) is finite for lattices of big rank but can contain elements of infinite order in small rank cases. We then study further two properties which appeared naturally in these investigations. A first which shows that property ND has a representation theoretical interpretation, while the other can be regarded as indicating that it might be hard to decide ND. Among others we show these two notions are equivalent for groups with SN.