arXiv subjects
Publications and source records attributed to Benjamin Sambale.
Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most 12. We extend their classification to dimension 13 and 14. As predicted by Donovan's Conjecture, we obtain only finitely many such Morita equivalence classes.
Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a theorem of Isaacs-Smith, as well as a recent result of three of the present authors.
We prove Brauer's k(B)-Conjecture for the 3-blocks with abelian defect groups of rank at most 5 and for all 3-blocks of defect at most 4. For this purpose we develop a computer algorithm to construct isotypies based on a method of Usami and Puig. This leads further to some previously unknown perfect isometries for the 5-blocks of defect 2. We also investigate basic sets which are compatible under the action of the inertial group.
This paper is an introduction and a survey to the concept of perfect isometries which was first introduced by Michel Brou{é} in 1990. Our main aim is to provide proofs of numerous results scattered in the literature. On the other hand, we make some observations which did not appear anywhere before.
Let $K:=\mathbb{Q}(G)$ be the number field generated by the complex character values of a finite group $G$. Let $\mathbb{Z}_K$ be the ring of integers of $K$. In this paper we investigate the suborder $\mathbb{Z}[G]$ of $\mathbb{Z}_K$ generated by the character values of $G$. We prove that every prime divisor of the order of the finite abelian group $\mathbb{Z}_K/\mathbb{Z}[G]$ divides $|G|$. Moreover, if $G$ is nilpotent, we show that the exponent of $\mathbb{Z}_K/\mathbb{Z}[G]$ is a proper divisor of $|G|$ unless $G=1$. We conjecture that this holds for arbitrary finite groups $G$.
We give a short and self-contained proof of a theorem of Ledermann and Neumann stating that there are only finitely many finite groups with a given number of automorphisms. We also discuss the history of related conjectures.
For a prime p, we call a positive integer n a Frobenius p-number if there exists a finite group with exactly n subgroups of order p^a for some $a\ge 0$. Extending previous results on Sylow's theorem, we prove in this paper that every Frobenius p-number $n\equiv 1\pmod{p^2}$ is a Sylow p-number, i.e., the number of Sylow p-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order 3^a for any $a\ge 0$.
One part of Sylow's famous theorem in group theory states that the number of Sylow p-subgroups of a finite group is always congruent to 1 modulo p. Conversely, Marshall Hall has shown that not every positive integer $n\equiv 1\pmod{p}$ occurs as the number of Sylow p-subgroups of some finite group. While Hall's proof relies on deep knowledge of modular representation theory, we show by elementary means that no finite group has exactly 35 Sylow 17-subgroups.
In a finite group G, we consider nilpotent weights, and prove a pi-version of the Alperin Weight Conjecture for certain pi-separable groups. This widely generalizes an earlier result by I. M. Isaacs and the first author.
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 number of irreducible characters in B. This is a variant of Brauer's Problem 21 and generalizes Külshammer's corresponding theorem for p-blocks of p-solvable groups. At the same time, our result generalizes Landau's classical theorem on the number of conjugacy classes of an arbitrary finite group. The proof relies on the classification of finite simple groups.
A well-known result of Scopes states that there are only finitely many Morita equivalence classes of $p$-blocks of symmetric groups with a given weight (or defect). In this note we investigate a lower bound on the number of those Morita equivalence classes.
We present a strong upper bound on the number k(B) of irreducible characters of a p-block B of a finite group G in terms of local invariants. More precisely, the bound depends on a chosen major B-subsection (u,b), its normalizer N_G(\langle u\rangle,b) in the fusion system and a weighted sum of the Cartan invariants of b. In this way we strengthen and unify previous bounds given by Brauer, Wada, Külshammer--Wada, Héthelyi--Külshammer--Sambale and the present author.
We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for $p$-elements, for $p$ from a prescribed set of primes.
Recently, Malle and Navarro put forward a projective version of Brauer's celebrated height zero conjecture on blocks of finite groups. In this short note we show that Brauer's original conjecture implies the projective version.
Answering a question of Pálfy and Pyber, we first prove the following extension of the k(GV)-Problem: Let G be a finite group and A\le Aut(G) such that (|G|,|A|)=1. Then the number of conjugacy classes of the semidirect product GA is at most |G|. As a consequence we verify Brauer's k(B)-Conjecture for pi-blocks of pi-separable groups which was proposed by Y. Liu. We also discuss equality in Brauer's Conjecture. On the other hand, we construct a counterexample to a version of Olsson's Conjecture for pi-blocks which was also introduced by Liu.
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.
Kiyota, Murai and Wada conjectured in 2002 that the largest eigenvalue of the Cartan matrix C of a block of a finite group is rational if and only if all eigenvalues of C are rational. We provide a counterexample to this conjecture and discuss related questions.
Let ZB be the center of a p-block B of a finite group with defect group D. We show that the Loewy length LL(ZB) of ZB is bounded by $\frac{|D|}{p}+p-1$ provided D is not cyclic. If D is non-abelian, we prove the stronger bound $LL(ZB)<\min\{p^{d-1},4p^{d-2}\}$ where $|D|=p^d$. Conversely, we classify the blocks B with $LL(ZB)\ge\min\{p^{d-1},4p^{d-2}\}$. This extends some results previously obtained by the present authors. Moreover, we characterize blocks with uniserial center.