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We consider an abelian group A which acts faithfully and coprimely on a solvable group G. We show that some A-orbit on Irr(G) must have cardinality divisible by almost half the primes in 7T(A). As a corollary, we improve a recent result of I. M. Isaacs concerning the maximum number of primes dividing any one character degree of a solvable group. A recent result of I. M. Isaacs [5, Corollary 4.3] relates the maximum number of primes dividing any one irreducible character degree of a solvable group G to the number of primes dividing all the character degrees of G taken together. Here we considerably strengthen the bound in [5, Corollary 4.3] by proving a result on character orbit sizes in coprime actions. We consider an abelian group A which acts faithfully and coprimely on a solvable group G. We show that some A-orbit on Irr(G) must have cardinality divisible by almost half the primes in 'n(A). Our approach roughly parallels that of [6]. This paper and [5] contain new applications of the results and methods of [6, 7, 1, 2]. I would like to thank I. M. Isaacs for bringing this problem to my attention. Our notation is largely standard. If a group G acts on a set Q and X E Q, we denote by OrbG(CO) the G-orbit of co. If G<iH and H also acts on Q, we say that h E H moves OrbG(CO) if Ch 4 OrbG(Co). All groups considered in this paper are finite and solvable. We now state our main results. THEOREM 1. Let A act faithfully on G with (I A IGI)= 1 and A abelian. Then 17T(A)I < 21I (OrbA(X))I + 8 forsome X E Irr(G). COROLLARY 1. Let G be solvable. Let s = max{I1T(x(l))I: X E Irr(G)} and let p = 17T(lX EIrr(G)X(l))L Then p < S2 + 1Os. PROOF. Using Theorem 1 above, we get a stronger version of [5, Theorem 4.1] in which part (a) is replaced by Ip(G) p(N) auI < 2(s Ia1) + 8. This leads to a stronger version of [5, Corollary 4.3] in which Ip(G)I < s + EZ-j(2(s i) + 8) = S2 + 10s. Since Ip(G) in [5] is called p in our paper, this completes the proof. Received by the editors November 13, 1985 and, in revised form, August 30, 1986. Presented on July 7, 1986 to the A.M.S. Summer Conference on Representation Theory of Finite Groups, Arcata, California. 1980 Mathematics Subject Classification (1985 Revision). Primary 20C15. ?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Primes dividing character degrees and character orbit sizes