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Fitting Height and Character Degree Graphs

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Abstract

Given a group G, Γ(G) is the graph whose vertices are the primes that divide the degree of some irreducible character and two vertices p and q are joined by an edge if pq divides the degree of some irreducible character of G. By a definition of Lewis, a graph Γ has bounded Fitting height if the Fitting height of any solvable group G with Γ(G)=Γ is bounded (in terms of Γ). In this note, we prove that there exists a universal constant C such that if Γ has bounded Fitting height and Γ(G)=Γ then h(G)≤C. This solves a problem raised by Lewis.

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Authors and Affiliations

  1. Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain

    Alexander Moretó

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  1. Alexander Moretó
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Correspondence to Alexander Moretó.

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Research supported by the Spanish Ministerio de Educación y Ciencia, MTM2004-06067-C02-01 and MTM2004-04665, the FEDER and Programa Ramón y Cajal.

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Moretó, A. Fitting Height and Character Degree Graphs. Algebr Represent Theor 10, 333–338 (2007). https://doi.org/10.1007/s10468-007-9047-4

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  • DOI: https://doi.org/10.1007/s10468-007-9047-4

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