Abstract. Let G be a finite solvable group, and let ∆(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P.P. Pálfy asserts that the complement∆(G) of the graph ∆(G) does not contain any cycle of length 3. In this paper we generalize Pálfy's result, showing that∆(G) does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of ∆(G) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if n is the clique number of ∆(G), then ∆(G) has at most 2n vertices. This confirms a conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for the famous ρ-σ conjecture by B. Huppert.

Abstract. Let G be a finite group acting faithfully on a finite vector space M in such a way that the centralizer of every element of M contains a Sylow q-subgroup of G as a central subgroup (for a fixed prime divisor q of jGj with ðq; jMjÞ ¼ 1). Then G is isomorphic to a subgroup of the semi-linear group on M.

Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of [1], where the solvable case was treated.

Let G be a finite group, and let ∆(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. It is well known that, whenever ∆(G) is connected, the diameter of ∆(G) is at most 3. In the present paper, we provide a description of the finite solvable groups for which the diameter of this graph attains the upper bound. This also enables us to confirm a couple of conjectures proposed by M.L. Lewis.

An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: