Sylow subgroups and the number of irreducible characters of degrees divisible by a prime p

2025/11/25 by Cossey, James P., Lewis, Mark L., Fry, A. A. Schaeffer +1
#20D10 #FOS: Mathematics #Group Theory (math.GR) #Primary 20C15 #Secondary 20D06

paper · doi:10.48550/arxiv.2511.20861

Abstract

Let G be a finite group and p a prime. We establish an upper bound for the derived length of a Sylow p-subgroup of G in terms of the number of irreducible characters of G whose degrees are divisible by p. We also prove that if B is a p-block of a finite p-solvable group G with defect group D, then the derived length of D is at most one more than the number of ordinary irreducible characters of positive height in B.

Citations

• On minimal positive heights for blocks of almost quasi-simple groups
• The Eaton-Moreto Conjecture and p-Solvable Groups
• Minimal heights and defect groups with two character degrees
• Brauer's Height Zero Conjecture
• Character degrees in blocks and defect groups

Related

• On the number of p'-degree characters in a finite group
• Products of characters and derived length
• On the characters of Sylow p-subgroups of finite Chevalley groups G(p^f) for arbitrary primes
• A note on restriction of Characters of Alternating groups to Sylow\n Subgroups
• Principal blocks with 5 irreducible characters