Abstract
We prove that a finite group G has a normal Sylow p-subgroup P if, and only if, every irreducible character of G appearing in the permutation character (1p)G with multiplicity coprime to p has degree coprime to p. This confirms a prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.
| Original language | English |
|---|---|
| Pages (from-to) | 552-567 |
| Number of pages | 16 |
| Journal | Bulletin of the London Mathematical Society |
| Volume | 54 |
| Issue number | 2 |
| Early online date | 14 Mar 2022 |
| DOIs | |
| Publication status | Published - Apr 2022 |
Bibliographical note
Acknowledgments:Part of this work was done while the fourth author was visiting the first at the University of Florence supported by the Spanish National Research Council through the ‘Ayuda extraordinaria a Centros de Excelencia Severo Ochoa’ (20205CEX001). The second author was supported by Emmanuel College, Cambridge. The third author was supported by ERC Consolidator Grant 647678. The fourth author was supported by the Spanish Ministerio de Ciencia e Innovación PID2019-103854GB-I00, PID2020-118193GA-I00 and FEDER funds. We thank Gabriel Navarro and Thomas Breuer for helping us checking Theorem 2.3 for some sporadic groups, and Gunter Malle for his comments on a previous version. Finally, we are grateful to the anonymous referee for accurate corrections that have improved the exposition of this paper.