proceedings of the american mathematical society Volume 107, Number I, September 1989 ON ABELIAN QUOTIENTS OF PRIMITIVE GROUPS MICHAEL ASCHBACHER AND ROBERT M. GURALNICK (Communicated by Warren J. Wong) Abstract. It is shown that if G is a primitive permutation group on a set of size n , then any abelian quotient of G has order at most n . This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if MjK is a minimal extension and L/K is an abelian extension contained in the normal closure of M, then the degree of L/K is at most the degree of M/K . 1. INTRODUCTION In this note, we prove the following results: Theorem 1. Let G be a primitive permutation group on a set of finite order n . Then \G: G'\<n. Theorem 2. Let G be a permutation group on a finite set of order n . Then |C?:fj'|<3"/3<2""1. Theorem 3. Let V be a finite dimensional vector space over a finite field of char- acteristic t and G a subgroup of GL(F). If Of(G) - I, then \G:G'\<\V\. Theorem 1 answers a question of Tamagawa, which was prompted by [I]. A minor modification of the argument in [I] shows that \G: G'(Hr\Hs)\ < \G: H\. This was first observed by D. Cantor in the following form: If a and ß are conjugate over a field K, then any abelian extension L/K with L c K(a,ß) satisfies \L: K\< \K(a): K\. The field theoretic version of Theorem 1 is: Corollary 4. Let K be a field with M/K a minimal extension. If L/K is an abelian extension with L contained in the normal closure of M, then \L: K\< \M:K\. Theorems 2 and 3 are needed to prove Theorem 1. Theorem 3 depends on the classification of simple groups (one needs to know the relative sizes of outer - Received by the editors November 10, 1988 and, in revised form, February 8, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20B15; Secondary 20C20, 20B05. Both authors were partially supported by NSF grants. The first author was partially supported by NSA. ©1989 American Mathematical Society 0002-9939/89 $1.00+$.25 per page 89 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use