On blocks with few irreducible characters

Duration: 30 mins 18 secs

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Description:	Noelia Rizo Universidad de Oviedo 09/05/2022 Programme: GR2W01 SemId: 35779


Created:	2022-05-18 10:22
Collection:	Groups, representations and applications: new perspectives
Publisher:	Isaac Newton Institute
Copyright:	Noelia Rizo
Language:	eng (English)


Abstract:	W. Burnside proposed to characterize finite groups with a given number of irreducible characters. The block-wise version of this problem is to characterize the defect groups of a -block of a finite group with a given number of irreducible characters in it. In this context, R. Brauer proved that if, and only if, the defect group of is trivial. Years later, J. Brandt showed that if, and only if, the defect groups of are cyclic of order 2. These two results do not require the Classification of Finite Simple Groups. However, the complexity of this problem seems to explode when we deal with the next situation, namely when we try to classify the defect groups of -blocks satisfying . It is conjectured that in this case the defect groups of are cyclic of order 3. When is the principal block or is normal in , the situation is much better understood and the conjecture is known to hold in this case. If and is the principal block, then S. Koshitani and T. Sakurai have proven that or 5. We show that the same is obtained whenever is an arbitrary block of and is normal in . Moreover, we deal with the next natural situation, namely where and is the principal block of , in which case we obtain that . This talk is an overview of joint works with J.M. Martínez, L. Sanus, M. Schaeffer Fry and C. Vallejo.

Abstract:

W. Burnside proposed to characterize finite groups with a given number of irreducible characters. The block-wise version of this problem is to characterize the defect groups of a -block of a finite group with a given number of irreducible characters in it.
In this context, R. Brauer proved that if, and only if, the defect group of is trivial. Years later, J. Brandt showed that if, and only if, the defect groups of are cyclic of order 2. These two results do not require the Classification of Finite Simple Groups. However, the complexity of this problem seems to explode when we deal with the next situation, namely when we try to classify the defect groups of -blocks satisfying . It is conjectured that in this case the defect groups of are cyclic of order 3. When is the principal block or is normal in , the situation is much better understood and the conjecture is known to hold in this case.
If and is the principal block, then S. Koshitani and T. Sakurai have proven that or 5. We show that the same is obtained whenever is an arbitrary block of and is normal in . Moreover, we deal with the next natural situation, namely where and is the principal block of , in which case we obtain that .
This talk is an overview of joint works with J.M. Martínez, L. Sanus, M. Schaeffer Fry and C. Vallejo.

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