  We describe an algorithm for obtaining generators of the unit group of the
integral group ring ZG of a finite abelian group G. We used our implementation
in Magma of this algorithm to compute the unit groups of ZG for G of order up
to 110. In particular for those cases we obtained the index of the group of
Hoechsmann units in the full unit group. At the end of the paper we describe an
algorithm for the more general problem of finding generators of an arithmetic
group corresponding to a diagonalizable algebraic group.
