Sylow branching coefficients and a conjecture of Malle and Navarro

We prove that a finite group G has a normal Sylow psubgroup 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.

[1]  Sylow branching coefficients for symmetric groups , 2019, Journal of the London Mathematical Society.

[2]  P. Py.  On the Representation Theory of the Symmetric Groups , 2005 .

[3]  Dock Bumpers,et al.  Volume 2 , 2005, Proceedings of the Ninth International Conference on Computer Supported Cooperative Work in Design, 2005..

[4]  G. Navarro,et al.  Characterizing normal Sylow p-subgroups by character degrees , 2012 .

[5]  P. Spiga,et al.  On the orders of zeros of irreducible characters , 2009 .

[6]  G. Navarro,et al.  McKay natural correspondences on characters , 2014 .

[7]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[8]  G. Michler.  Brauer's conjectures and the classification of finite simple groups , 1986 .

[9]  G. Navarro,et al.  Brauer’s Height Zero Conjecture for principal blocks , 2021, 2102.08270.

[10]  E. Giannelli,et al.  On permutation characters and Sylow p-subgroups of Sn , 2018, Journal of Algebra.

[11]  B. M. Fulk.  MATH , 1992 .

[12]  D. Passman,et al.  Character Theory of Finite Groups , 2010 .

[13]  Restricting irreducible characters to Sylow -subgroups , 2018 .

[14]  G. Navarro,et al.  Restriction of odd degree characters and natural correspondences , 2016, 1601.04423.

[15]  Andrew Granville,et al.  Defect zero p-blocks for finite simple groups , 1996 .

[16]  N. Ito.  Some Studies on Group Characters , 1951, Nagoya Mathematical Journal.

[17]  Jeremy Sylvestre,et al.  Representations of Finite Groups , 2015, Arch. Formal Proofs.

[18]  J. B. Olsson.  On the p-blocks of symmetric and alternating groups and their covering groups , 1990 .

[19]  G. James,et al.  The Representation Theory of the Symmetric Group , 2009 .

[20]  Pham Huu Tiep,et al.  Character restrictions and multiplicities in symmetric groups , 2017 .

[21]  Jorn B. Olson.  McKay numbers and heights of characters. , 1976 .

[22]  Character sums and double cosets , 2008 .

[23]  G. Navarro.  Character Theory and the McKay Conjecture , 2018 .

[24]  Jørn B. Olsson,et al.  Combinatorics and representations of finite groups , 1993 .