Related papers: Principal $2$-Blocks and Sylow $2$-Subgroups

Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise…

Let $G\subset\GL(\BC^r)$ be a finite complex reflection group. We show that when $G$ is irreducible, apart from the exception $G=\Sgot_6$, as well as for a large class of non-irreducible groups, any automorphism of $G$ is the product of a…

In the representation theory of finite groups, Brou\'e's abelian defect group conjecture says that for any prime p if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the…

We study Brauer's long-standing $k(B)$-conjecture on the number of characters in $p$-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for $p\ge5$ nor in the case of abelian defect.…

Let $P$ be a Sylow $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime. Using a normal $p$-series $\mathcal{N}$ of $G$, we introduce the notion of $(\mathcal{N},p)$-stable characters and prove that $G$ and ${\bf N}_G(P)$…

M. Kiyota, T. Okuyama and T. Wada recently proved that each 2-block of a finite symmetric group contains a unique irreducible Brauer character that has height 0. We present a more conceptual proof of this result.

We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets…

Slattery has generalized Brauer's theory of p-blocks of finite groups to pi-blocks of pi-separable groups where pi is a set of primes. In this setting we show that the order of a defect group of a pi-block B is bounded in terms of the…

In this paper, we construct a group with three real irreducible characters whose Sylow 2-subgroup is an iterated central extension of a Suzuki 2-group. This answers a question raised by Moreto and Navarro who asked whether such a group…

Many results have been established about determining whether or not an element evaluates to zero on an irreducible character of a group. In this note it is shown that if a group $G$ has a normal nilpotent subgroup $N$, and $P$ is a Sylow…

We study the restriction to Sylow subgroups of irreducible characters of symmetric groups. In particular, we focus our attention on constituents of degree greater than 1. Our main result is a wide generalization of Theorem 3.1 of Giannelli…

Eaton and Moret\'o proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We…

This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the…

Let $G$ be a finite group and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. We conjecture that there exists $x \in G$ such that $P_i \cap P_i^x$ is inclusion-minimal in $\{ P_i \cap P_i^g : g \in G\}$ for all…

The Malle-Navarro conjecture relates central block theoretic invariants in two inequalities. In this paper, we prove the conjecture for the 2-blocks and the unipotent 3-blocks of the general linear and unitary groups in non-defining…

Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence,…

Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a…

The celebrated It\^o-Michler theorem asserts that a prime $p$ does not divide the degree of any irreducible character of a finite group $G$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup. The principal block case of the…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

If a finite group $A$ acts coprimely as automorphisms on a finite group $G$, then the $A$-invariant Brauer $p$-blocks of $G$ are exactly those that contain $A$-invariant irreducible characters.