On principal indecomposable degrees and Sylow subgroups

Abstract

We conjectured in Malle and Navarro (J Algebra 370:402–406, 2012) that a Sylow p-subgroup P of a finite group G is normal if and only if whenever p does not divide the multiplicity of \(\chi \in {{\text {Irr}}}(G)\) in the permutation character \((1_P)^G\), then p does not divide the degree \(\chi (1)\). In this note, we prove an analogue of this for p-Brauer characters.

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Correspondence to Gabriel Navarro.

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Gunter Malle gratefully acknowledges financial support by SFB TRR 195. He thanks the Isaacs Newton Institute for Mathematical Sciences in Cambridge for support and hospitality during the programme “Groups, Representations and Applications: New Perspectives” when work on this paper was undertaken. This work was supported by: EPSRC grant number EP/R014604/1. The research of Gabriel Navarro is supported by Ministerio de Ciencia e Innovación PID2019-103854GB-I00 and FEDER funds.

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Malle, G., Navarro, G. On principal indecomposable degrees and Sylow subgroups. Arch. Math. (2020). https://doi.org/10.1007/s00013-020-01494-9

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Keywords

  • Normal Sylow subgroups
  • Dimensions of projective indecomposable modules
  • Brauer character degrees

Mathematics Subject Classification

  • 20C20