Integral group rings of solvable groups with trivial central units

Abstract

Abstract The integral group ring {\mathbb{Z}G} of a group G has only trivial central units if the only central units of {\mathbb{Z}G} are {\pm z} for z in the center of G. We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.