On principal indecomposable degrees and Sylow subgroups
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Archiv der Mathematik
On principal indecomposable degrees and Sylow subgroups Gunter Malle and Gabriel Navarro
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 χ ∈ Irr(G) in the permutation character (1P )G , then p does not divide the degree χ(1). In this note, we prove an analogue of this for p-Brauer characters. Mathematics Subject Classification. 20C20. Keywords. Normal Sylow subgroups, Dimensions of projective indecomposable modules, Brauer character degrees.
1. Introduction. In [3], we proved that if G is a finite group and P ∈ Sylp (G), then P G if and only if p does not divide the degrees of the irreducible constituents of the permutation character (1P )G . Furthermore, we conjectured that this happens if and only if the irreducible constituents of (1P )G with multiplicity not divisible by p had degree not divisible by p. This has turned out to be a difficult problem for symmetric groups (in the cases where p = 2 or p = 3). In this short note, we aim for an analogous result on modular characters. By using the induction formula and [5, Theorem 2.13], notice that the permutation p-Brauer character ((1P )G )0 decomposes as Φϕ (1) ((1P )G )0 = ϕ, |G|p ϕ∈IBr(G)
where Φϕ is the projective indecomposable character associated with the irreducible Brauer character ϕ. 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 Innovaci´ on PID2019-103854GB-I00 and FEDER funds.
G. Malle and G. Navarro
Arch. Math.
We consider the following modular analogue of our conjecture in [3]: Theorem A. Let p be a prime, let G be a finite group, and let P ∈ Sylp (G). Then P G if and only if whenever ϕ ∈ IBr(G) is such that Φϕ (1)/|P | is not divisible by p, then ϕ(1) is not divisible by p. This generalizes the well-known result of G. Michler [4, Theorem 5.5] that G has a normal Sylow p-subgroup if and only if all irreducible Brauer characters of G have degree prime to p. Our proof (as in [4]) relies on the classification of finite simple groups for p odd only. 2. Proof of Theorem A. We use the notation in [5]. Let p be a fixed prime. We choose a maximal ideal of the ring of algebraic integers in C containing p, with respect to which we calculate the set IBr(G) of irreducible (p-)Brauer characters for every finite group G. If N G and θ ∈ IBr(N ), then IBr(G|θ) is the set of irreducible constituents of the induced character θG , which, by [5, Corollary 8.7], is the set of irreducible Brauer characters ϕ ∈ IBr(G) such that θ is a constituent of the restriction ϕN . For ϕ ∈ IBr(G), we write cϕ = Φϕ (1)/|G|p . (This is a
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