On Height-Zero Characters in p-Constrained Groups

Abstract

Consider G to be a finite group and p to be a prime divisor of the order |G| in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro’s vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ∈Irr(G) have a positive height, and let there be an anchor group Aϕ. We prove that if the normalizer NG(Aϕ) is p-constrained, then Op´(NG(Aϕ))≠{1G}, where Op´(NG(Aϕ)) is the maximal normal p´ subgroup of NG(Aϕ). We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results.