Let G be a finite group. Attach to G the following graph Γ: its vertices are the non‐central conjugacy classes of G, and two vertices are connected if their cardinalities are not coprime. Denote by n(Γ) the number of the connecte components of Γ. We prove that n(Γ) ⩽ 2 for all finite groups, and we completely characterize groups with n(Γ) = 2. When Γ is connected, then the diameter of the graph is at most 4. For simple non‐abelian finite groups, the graph is complete. Similar results are proved for infinite FC‐groups.

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If S is a subset of a group G, we define its square S^2 by the formula S^2 = {ab | a, b ∈S}. We prove that if S is a finite subset of an ordered group that generates a nonabelian group, then\ud the order of S^2 is bigger or equal to 3|S|-2. This generalizes a classical result from the theory of set addition. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INDAM

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