Actions and invariant character degrees. II (Q1922887)

In part I [ibid. 160, No. 1, 172-178 (1993; Zbl 0805.20007)] the author gave evidence that there exists a rich and useful relationship between the \(A\)-stable structure of a finite group \(G\) on which \(A\) acts coprimely, and the set \(\text{cd}_A(G)\) of degrees of the complex irreducible \(A\)-invariant characters of \(G\). What can be said about the fixed-points subgroup \(C_G(A)\) if much is known about \(\text{cd}_A(G)\)? In this paper some theory is developed centering around the above theme. The following is the main outcome: Theorem A. Suppose that a finite group \(A\) acts coprimely on a finite supersolvable group \(G\). Then the derived length of \(C_G(A)\) is bounded by \(|\text{cd}_A(G)|\).