The number of composite character degrees in solvable groups

Abstract

ABSTRACTLet cd(G) be the set of irreducible character degrees of a finite group G. Also, let cdP(G)⊆cd(G) and cdC(G)⊆cd(G) be the sets of all prime character degrees and all composite character degrees of G, respectively. We prove that for a solvable group G, if |cdC(G)|≥4, then . This extends the results of [4], [7], [2] and [3] where it has been proved that |cdP(G)|≤3 if |cdC(G)| = 0,1,or2, and |cdP(G)|≤4 if |cdC(G)| = 3, respectively.