Character degrees, blocks and defect groups

• Autores: Josep Miquel Martínez Marín
• Directores de la Tesis: Lucía Sanus Vitoria (dir. tes.) , Gabriel Navarro Ortega (codir. tes.)
• Lectura: En la Universitat de València ( España ) en 2023
• Idioma: inglés
• Número de páginas: 124
• Tribunal Calificador de la Tesis: María José Felipe (presid.) , Alexander Moretó Quintana (secret.) , Silvio Dolfi (voc.)
• Enlaces
  • Tesis en acceso abierto en:  webges.uv.es (pdf)  RODERIC 
• Resumen

  • This thesis explores connections between properties of p-blocks and their defect groups, related to Brauer's problem 21, the Alperin-McKay conjecture and a recent question of G. Navarro.

    In chapter 1 we give prelminaries on characters and blocks of finite groups.

    In chapter 2 we prove that if a p-block B contains exactly four irreducible characters, then its defect groups must be of order 4 or 5, assuming that the Alperin-McKay conjecture holds for B. This is done by proving the result for blocks with normal defect group, which ends up requiring a study of primitive permutation groups of low rank and their projective characters via results of P. Fong and W. F. Reynolds and ordinary- modular character triples. This result contributes to Brauer's problem 21 by expanding the classification of defect groups for blocks with a fixed number of irreducible characters. This is joint work with N. Rizo and L. Sanus.

    In chapter 3 we study a recent question of Navarro which asks whether the number of distinct degrees of irreducible characters in a block bounds the derived length of its defect groups. This is connected to a Conjecture of A. Moretó which asks whether this derived length can be bounded in terms of the number of distinct heights of characters in the block. We prove Navarro's conjecture for principal blocks with up to 3 character degrees. The main results are proved via a reduction to simple groups, and some proofs for simple groups are included. Further, we prove a version of Thompson's theorem on character degrees for principal blocks, which is much more elementary in natura and does not require the classification of finite simple groups. Part of these results were joint work with E. Giannelli and A. A. Schaeffer Fry.

    In chapter 4 we prove that, for p-solvable groups, there are bijections confirming the Alperin-McKay conjecture with degree divisibility. This extends previous results of A. Turull and N. Rizo for the McKay conjecture. Its proof requires deep results due to Turull on the Glauberman correspondence and some results of M. Murai on the behavior of the Gallagher correspondence and blocks. This also provides interesting consequences for nilpotent blocks and the Dade-Glauberman-Nagao correspondence in p-solvable groups. Further, our techniques allow us to prove that the dimension of block is divisible by the dimension of its Brauer correspondent in p-solvable groups. This is joint work with D. Rossi.