Even-order subgroups of finite-dimensional division rings

Abstract

Let K be a field. A finite group G is called K-adequate if there exists a K-division ring D (a finite-dimensional division algebra central over K) such that G is contained in D*, the multiplicative group of nonzero elements of D. Fein and Schacher have investigated the problem of determining for which fields K there exists a noncyclic group of odd order which is K-adequate. In [5, 61 they solved this problem when K is an algebraic number field or p-local field. In this paper we do the same for noncyclic groups of even order. Our main result is that if K is an algebraic number field then there exists a noncyclic group of even order which is K-adequate. We show this is false for p-local fields and determine necessary and sufficient conditions on K for there to exist an even-order noncyclic group which is K-adequate. We adopt the notation and terminology of [6]. Recall that for any natural number TZ, cIZ denotes a primitive nth root of unity. If u and ZI are integers j3(u, V) is the highest power of u dividing v and [u, v] is the order of u modulo v. If K is an algebraic number field and y is a prime of K we denote the completion of K at y by K,, . For a finite extension L of K and a prime 7 of L dividing y we denote the relative degree of q over y by f(~/r). If E and L are local fields we denote the ramification degree from L to E by e(E/L). We assume all algebras are finite-dimensional over their centers and all fields are of characteristic zero. We use freely the classification of division algebras over global and local fields by means of Hasse invariants. The reader is referred to [3, Chap. 71 for a discussion of this material.