On the Number of Components of a Graph Related to Character Degrees

Abstract

ABSTRACT. We connect two nonlinear irreducible character of a finite group G if their degrees have a common prime divisor. In this paper we show that the corresponding graph has at most three connected components.

Primes dividing the degrees of the real characters

“…We will largely follow the ideas of [22, p. 33-35] to show that for any odd prime p ≤ n, S has a real irreducible character χ such that 2 p|χ(1). As in [22], we choose χ to be an irreducible constituent of ξ | S , where ξ is the irreducible character of Sym n corresponding to the partition α = (n − r − s, s + 1, 1 r −1 ) with 0 ≤ r − 1, s, r + 2s + 1 ≤ n. In particular,…”

“…In what follows, the choices of α are made following [22], and necessary divisibility properties of ξ(1) were proved therein. For the reader's convenience, we will recall these choices here.…”

Diameters of degree graphs of nonsolvable groups

“…In [9], it is shown that for n 7 and n = 8, if p is an odd prime dividing the order of A n , then A n has an irreducible character χ such that 2p | χ (1). Thus every odd prime in ρ(G) is adjacent to 2, so the diameter of ∆(G) is at most 2.…”

Hamiltonian character graphs

“…One of these graphs is the character graph Δ(G) of G. Its vertex set is ρ(G) and two vertices p and q are joined by an edge if the product pq divides some character degree of G. This graph was first defined in [9] and has been studied extensively since then. We refer the readers to a survey by Lewis [3] for results concerning this graph and related topics.…”