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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.

Abstract.Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others. The purpose of this article is to consider the other extreme, namely, to characterize all finite groups in which all the nonlinear irreducible characters have distinct degrees.Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others (see [8, Chapter 12]). The purpose of this article is to consider the other extreme, namely, the case when all nonlinear irreducible character degrees are distinct. We prove Theorem. Let G be a nonabelian finite group. Let {d\, 82, ... , 6r} be the set of all nonlinear irreducible ordinary characters of G. Assume that 6¡( 1 ) / 6}■ ( 1 ) for all i / j. Then one of the following holds:( 1 ) G is an extra-special 2-group. (2) G is a Frobenius group of order pn(pn -1) for some prime power pn with an abelian Frobenius kernel of order p" and a cyclic Frobenius complement. (3) G is a Frobenius group of order 72 in which the Frobenius complement is isomorphic to the quaternion group of order 8.Remarks. All the groups described above satisfy the assumption of the theorem. The groups of type (1) and (2) have exactly one nonlinear character degree, while the group in (3) has two such character degrees. To show that there are no perfect groups satisfying the assumption, we use the classification of the finite simple groups. The proof for nonperfect groups is independent of the classification.In [10], Seitz shows that if in the theorem r = 1 , then either (1) or (2) holds.Notation. Most of our notation is standard and taken mainly from [8]. We will denote the set of all irreducible ordinary characters of the finite group G by

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