Search results for "SYLOW"

Sylow subgroups, exponents, and character values

2019

If G G is a finite group, p p is a prime, and P P is a Sylow p p -subgroup of G G , we study how the exponent of the abelian group P / P ′ P/P’ is affected and how it affects the values of the complex characters of G G . This is related to Brauer’s Problem 12 12 . Exactly how this is done is one of the last unsolved consequences of the McKay–Galois conjecture.

Rationality and Sylow 2-subgroups

2010

AbstractLet G be a finite group. If G has a cyclic Sylow 2-subgroup, then G has the same number of irreducible rational-valued characters as of rational conjugacy classes. These numbers need not be the same even if G has Klein Sylow 2-subgroups and a normal 2-complement.

On finite minimal non-nilpotent groups

2005

[EN] A critical group for a class of groups X is a minimal non-X-group. The critical groups are determined for various classes of finite groups. As a consequence, a classification of the minimal non-nilpotent groups (also called Schmidt groups) is given, together with a complete proof of Gol¿fand¿s theorem on maximal Schmidt groups.

On the blockwise modular isomorphism problem

2017

As a generalization of the modular isomorphism problem we study the behavior of defect groups under Morita equivalence of blocks of finite groups over algebraically closed fields of positive characteristic. We prove that the Morita equivalence class of a block B of defect at most 3 determines the defect groups of B up to isomorphism. In characteristic 0 we prove similar results for metacyclic defect groups and 2-blocks of defect 4. In the second part of the paper we investigate the situation for p-solvable groups G. Among other results we show that the group algebra of G itself determines if G has abelian Sylow p-subgroups.

A generalization to Sylow permutability of pronormal subgroups of finite groups

2020

[EN] In this note, we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterized in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterized in terms of pronormality.

Characters and generation of Sylow 2-subgroups

2021

Sylow Normalizers with a Normal Sylow 2-Subgroup

2008

AbstractIf G is a finite solvable group and p is a prime, then the normalizer of a Sylow p-subgroup has a normal Sylow 2-subgroup if and only if all non-trivial irreducible real 2-Brauer characters of G have degree divisible by p.

Restricting irreducible characters to Sylow 𝑝-subgroups

2018

We restrict irreducible characters of finite groups of degree divisible by p p to their Sylow p p -subgroups and study the number of linear constituents.

Derived length and character degrees of solvable groups

2003

We prove that the derived length of a solvable group is bounded in terms of certain invariants associated to the set of character degrees and improve some of the known bounds. We also bound the derived length of a Sylow p-subgroup of a solvable group by the number of different p-parts of the character degrees of the whole group.

Degrees of characters in the principal block

2021

Abstract Let G be a finite group. We prove that if the set of degrees of characters in the principal p-block of G has size at most 2 then G is p-solvable, and G / O p ′ ( G ) has a metabelian normal Sylow p-subgroup. The general question of proving that if an arbitrary p-block has two degrees then their defect groups are metabelian remains open.