On linear relations between character values

AbstractLet s and u be nonconjugate elements of a finite group and let a, b, and c be complex numbers, ab ≠ 0. Groups satisfying aX(s) + bX(u) = c for every non-principal irreducible character X are completely classified. They are the PSL(2, 2n), n ⩾ 2 and some solvable group of derived length at most 3

Finite groups with almost distinct character degrees

AbstractFinite groups with the nonlinear irreducible characters of distinct degrees, were classified by the authors and Berkovich. These groups are clearly of even order. In groups of odd order, every irreducible character degree occurs at least twice. In this article we classify finite nonperfect groups G, such that χ(1)=θ(1) if and only if θ=χ¯ for any nonlinear χ≠θ∈Irr(G). We also present a description of finite groups in which xG′⊆class(x)∪class(x−1) for every x∈G−G′. These groups generalize the Frobenius groups with an abelian complement, and their description is needed for the proof of the above mentioned result on characters

Covering the alternating groups by products of cycle classes

AbstractGiven integers k,l⩾2, where either l is odd or k is even, we denote by n=n(k,l) the largest integer such that each element of An is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(An,Cl), where Cl is the set of all l-cycles in An. We prove that if k⩾2 and l⩾9 is odd and divisible by 3, then 23kl⩽n(k,l)⩽23kl+1. This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368–380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87–99]

On groups with average element orders equal to the average element order of the alternating group of degree (5)

Let (G) be a finite group. Denote by (psi(G)) the sum (psi(G)=sum_{xin G}|x|,) where (|x|) denotes the order of the element (x), and by (o(G)) the average element orders, i.e. the quotient (o(G)=frac{psi(G)}{|G|}.) We prove that (o(G) = o(A_5)) if and only if (G simeq A_5), where (A_5) is the alternating group of degree (5)