Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3

Abstract

If G is a finite group whose integral group ring $\mathbb{Z}[G]$ has the multiplicative Jordan decomposition property, then it is known that all Wedderburn components of the rational group ring $\mathbb{Q}[G]$ have degree at most 3. While degree 3 components can occur, we prove here that if they do, then certain central units in $\mathbb{Z}[G]$ cannot exist. With this, we are able to greatly simplify the argument that characterizes those 3-groups with integral group ring having MJD. Furthermore, we show that if G is a nonabelian semidirect product of the form $C_p⋊C_{3^k}$, with prime $p>7$ and with the cyclic 3-group acting like a group of order 3, then $\mathbb{Z}[G]$ does not have MJD.