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

Computing generators of the unit group of an integral abelian group ring