If is an irreducible character of a finite group G, then the codegree of is jG W ker. /j= .1/. We show that if G is a p-group, then the nilpotence class of G is bounded in terms of the largest codegree for an irreducible character of G.
Abstract. Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) − {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.
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