\(SK_1\) of finite Abelian groups. I
For many years the Whitehead group of a group \(G\) had resisted computation. In the finite case, the portion that was so difficult to compute was \(SK_1(\mathbb{Z}[G])\). In this paper, it is shown that \[ SK_1(\mathbb{Z}[G])\simeq \prod_{p/| G|, p\neq 2} SK_1(\mathbb{Z}[G_p])^{q(G/G_p)} \times SK_1(\mathbb{Z}[G_2]) \times SK_1(\mathcal O[G_2])^{q(G/G_2)-1} \] where \(G_p\) is the Sylow \(p\)-subgroup of the finite group \(G\), \(q(H)\) is the number of irreducible irrational representations of a group \(H\), and \(\mathcal O\) is any totally imaginary ring of algebraic integers in which \(2\) is totally unramified. In the sequel [the authors et al., Invent. Math. 87, 253--302 (1987; Zbl 0605.18006)] \[ SK_1(\mathbb{Z}[G])\simeq \prod_{p/| G|}SK_1(\mathbb{Z}[G_p])^{q(G/G_p)} \times (\mathbb{Z}/2)^{(q(G/G_p)-1)(2^n-1-n-\binom{n}{2})}, \] where \(n\) is the order of \(G\). An efficient algorithm and many explicit computations are also given in the sequel.
totally imaginary ring of algebraic integers
scientific article; zbMATH DE number 3920756