\(SK_1\) of finite abelian groups. II (Q1084166)

This paper is, as advertised, a continuation of [Invent. Math. 82, 1--18 (1985; Zbl 0576.18007)]. In the first paper there where complications whenever \(2\)-groups were involved. These complications are removed in this paper. In particular, if \(H\) is an abelian \(2\)-group and \(\mathfrak O\) is a totally imaginary ring of integers in which \(2\) is unramified, then \(SK_1(\mathfrak O[H])\) is computed up to \(SK_1(\mathbb{Z}[H]) \). The formulae of part I are simplified in part II and the simplified description is used to compute \(SK_1(\mathbb{Z}[G])\) for elementary abelian \(p\)-groups. In part I it was shown that \(SK_1(\mathfrak O[G])\) is zero if \(G\) is cyclic and the primes dividing the order of \(G\) do not ramify in \(\mathfrak O\). In this paper the restriction due to ramification is removed and the result is generalized, namely: For any ring of algebraic integers, \(SK_1(\mathfrak O[G])=0\) if \(G\) is metacyclic. Conversely, if \(G\) is a finite group and \(SK_1(\mathfrak O[G])=0\) for all rings of algebraic integers \(\mathfrak O\), then \(G\) is metacyclic. The paper concludes with the computation of the exponent of \(SK_1(\mathbb{Z}[G])\) for any finite abelian group \(G\) and the computation of some examples of \(SK_1(\mathbb{Z}[G])\) for various explicit \(G\).