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Writing units of integral group rings of finite abelian groups as a product of Bass units

Journal Contribution - Journal Article

We give a constructive proof of a theorem of Bass and Milnor saying that if $G$ is a finite abelian group then the Bass units of the integral group ring $\Z G$ generate a subgroup of finite index in its units group $\U(\Z G)$. Our proof provides algorithms to represent some units that contribute to only one simple component of $\Q G$ and generate a subgroup of finite index in $\U(\Z G)$ as product of Bass units.
We also obtain a basis $B$ formed by Bass units of a free abelian subgroup of finite index in $\U(\Z G)$ and give, for an arbitrary Bass unit $b$, an algorithm to express $b^{\varphi(|G|)}$ as a product of a trivial unit and powers of at most two units in this basis $B$.
Journal: Mathematics of Computation
ISSN: 0025-5718
Volume: 83
Pages: 461-473
Publication year:2014
Keywords:Integral group rings, finite abelian groups, units
  • ORCID: /0000-0002-2695-7949/work/70477321
  • Scopus Id: 84888017499