Related papers: Blocks with transitive fusion systems

Let $p$ be an odd prime and $S$ a nonabelian finite $p$-group. In [9, 10], they proposed the following conjecture: if $\mathcal{F}$ be a transitive fusion system over a finite $p$-group $S$, then $S$ is either extraspecial of order $p^{3}$…

We prove that if $B$ is a $p$-block with non-trivial defect group $D$ of a finite $p$-solvable group $G$, then $\ell(B) < p^r$, where $r$ is the sectional rank of $D$. We remark that there are infinitely many $p$-blocks $B$ with non-Abelian…

Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on…

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety…

In this paper, we find some exotic fusion systems which have non-trivial strongly closed subgroups, and we prove these fusion systems are also not realizable by p-blocks of finite groups.

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

Given a prime $p$, we construct a permutation group containing at least $p^{p-2}$ non-conjugated regular elementary abelian subgroups of order $p^3$. This gives the first example of a permutation group with exponentially many non-conjugated…

Let $p$ be a prime. In this paper, we compute complexities of some simple modules of symmetric groups labelled by two-part partitions. Most of the simple modules considered here are contained in the $p$-blocks with non-abelian defect…

In the representation theory of finite groups, there is a well-known and important conjecture due to M. Brou\'e. He conjectures 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…

We prove the Alperin-McKay Conjecture for all $p$-blocks of finite groups with metacyclic, minimal non-abelian defect groups. These are precisely the metacyclic groups whose derived subgroup have order $p$. In the special case $p=3$, we…

We prove that if $b$ is a block of a finite group with normal abelian defect group and inertial quotient a direct product of elementary abelian groups, then $\operatorname{Picent}(b)$ is trivial. We also provide examples of blocks $b$ of…

Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we…

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander-Reiten quiver of the principal block of a finite group with non-cyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite…

Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2. Suppose that B has an elementary abelian defect group of order 16 and only one simple module. The main…

We compare lower defect groups associated with $p$-regular classes and vertices of simple modules for a block of a finite group algebra. We show that lower defect groups are contained in vertices of simple modules after suitable reordering.…

Let $p$ be a prime number. We compute the trivial source character tables of finite Frobenius groups $G$ with an abelian Frobenius complement $H$ and an elementary abelian Frobenius kernel of order $p^2$. More precisely, we deal with all…

Building on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of $\mathbb{Z}_p G$ of defect 1. This allows us to prove that if $p$ is a prime and $G$ is a…

In this paper, we prove that a \(p\)-block with abelian defect group is inertial if it covers a \(p\)-block of a normal subgroup of \(p\)-power index having only one irreducible Brauer character orbit.

Elements $a,b$ of a semigroup $S$ are said to be \emph{primarily conjugate} or just \emph{p-conjugate}, if there exist $x,y\in S^1$ such that $a=xy$ and $b=yx$. The p-conjugacy relation generalizes conjugacy in groups, but for general…

In this paper, I show that if $p$ is an odd prime, and if $P$ is a finite $p$-group, then there exists an exact sequence of abelian groups $$0\to T(P)\to D(P)\to\lproj{P}\to H^1\big(\apdeux(P),\Z\big)^{(P)},$$ where $D(P)$ is the Dade group…