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Units in integral cyclic group rings for order LRPS Ferguson, Ronald Aubrey

Abstract

For a finite abelian group A, the group of units in the integral group ring ZA may be written as the direct product of its torsion units ±A with a free group U2A . Of finite index in U2A is the group ΩA, the elements of U2 which are mapped to cyclotomic units by each character of A. The order of U2A/ΩA depends on class numbers [formula] in real cyclotomic rings [formula]. Of finite index in ΩA is the group of constructible units YA, for which a multiplicative basis may be explicitly written. The order of ΩA/YA is the circular index c(A). In many cases, for example where A is a p-group with p a regular prime, this index is trivial. This thesis develops an inductive theory for determining c(Cn) where Cn is a cyclic group of order n = lrps, with I and p distinct primes, and also for giving some description of the group ΩCnjYCn. This is a continuation of the work of Hoechsmann for the case n = Ip. It turns out that the methods required for n = lrps are, in general, very different from the ones used for r = s = 1.

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