A class of soluble groups

Abstract

In this paper a problem in group theory is solved to produce an interesting class of groups. The motivation, however, comes from algebraic number theory and in particular the Artin representation. To describe this representation we follow [5, Chap. IV and VI]. Let K and L be local fields, i.e., fields complete with respect to a discrete valuation and with perfect residue field. Let L have valuation ring A, and valuation v, extending the valuation on K. Suppose that L/K is a finite Galois extension with Galois group H. Then H has a normal series consisting of the ramification groups Hj = {O E H: V~(UX X) > j + 1 V x E A,}, j = -1, 0, I,2 ,..., which produces a character of H, due to Artin, as described in Theorem 1 below. The notation adopted in this paper is as follows. If G is a finite group, then Irr(G) denotes its set of irreducible characters. l,, rG, and U, = rc 1, are its principal, regular, and augmentation characters, respectively, whilst unless otherwise stated xc and xG will mean the restriction of x to G and the character of G induced from x.