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The integral group ringℤ  ⁢  G{\mathbb{Z}G}    of a group  G  has only trivial central units if the only central units ofℤ  ⁢  G{\mathbb{Z}G}    are±  z{\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.

Integral group rings of solvable groups with trivial central units