arXiv subjects
Publications and source records attributed to Alexander Moretó.
Fix an integer $m$ bigger than 2. We prove that if there exists a finite group with $mp+1$ Suylow $p$-subgroups, where $p$ is large enough, then $mp+1$ is prime. This improves on a theorem of M. Hall and is a partial answer to Brauer's Problem 26. Our proof uses techniques from analytic number theory, and it also raises new questions in that area.
This paper proposes a conjectural framework for Alperin's Main Problem of Block Theory from 1976. The character sets considered here are defined by nonvanishing at given elements, not only by degree conditions. From this point of view, McKay's conjecture is usually recovered as a first degree-level consequence. The guiding idea is that the right local objects governing character values are not, in general, the sets ${\rm Irr}_{p'}(G)$ and the normalizers of Sylow $p$-subgroups, but rather the sets ${\rm Irr}^x(G)$ of irreducible characters not vanishing at a given element $x$, together with the subnormalizer subgroup ${\rm Sub}_G(x)$. I state the basic conjectures of this theory, propose stronger versions, and verify the main conjectures in several families, including the simple groups with TI Sylow $p$-subgroups. I also show how this perspective reorganizes several classical questions in character theory.
This article is essentially an English translation of a paper of mine, published in \emph{La Gaceta de la RSME}. Its aim is to present, for a broad mathematical audience, a research programme in local representation theory that goes beyond the classical restrictions to characters of $p'$-degree, characters of height zero, and blocks of abelian defect. The final and most recent part of this programme concerns Alperin's main problem of block theory: the search for local rules for character values. In that direction I describe the conjectures on picky elements and subnormalizers, which suggest that the sets ${\rm Irr}^x(G)$ and the subgroups ${\rm Sub}_G(x)$ are the natural objects attached to a $p$-element $x$.
Let $G$ be a finite group and let $p$ be a prime. In this paper, we prove a strengthened version of Brauer's height zero conjecture for the principal $p$-block of $G$ that takes the action of a certain group of Galois automorphisms into account. This answers a conjecture recently proposed by Malle, Moretó, Rizo and Schaeffer Fry. We then use this to obtain a structural result which can be seen as a Galois version of the Itô-Michler theorem.
Let $G_p$ be the set of $p$-elements of a finite group $G$. Do we need all the Sylow $p$-subgroups of $G$ to cover $G_p$? Although this question does not have an affirmative answer in general, our work indicates that the answer is yes more often than one could perhaps expect.
A new conjecture on characters of finite groups, related to the McKay conjecture, was proposed recently by the first and third authors. In this paper, we prove it for $p$-solvable groups when $p$ is odd.
Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.
The celebrated Itô-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 recently-proven Brauer's height zero conjecture isolates the abelian part in the Itô-Michler theorem. In this paper, we show that the normal part can also be isolated in a similar way. This is a consequence of work on a strong form of the so-called Brauer's height zero conjecture for two primes of Malle and Navarro. Using our techniques, we also provide an alternate proof of this conjecture.
Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts that if $G$ is a finite group with the same set of complex character degrees as $H$, then $G\cong H\times A$ for some abelian group $A$. Over the past two decades, several specific cases of this conjecture have been addressed. Recently, attention has shifted to the analogous conjecture for character codegrees: if $G$ has the same set of character codegrees as $H$, then $G\cong H$. Unfortunately, both problems have primarily been examined on a case-by-case basis. In this paper and the companion [HM22], we present a more unified approach to the codegree conjecture and confirm it for several families of simple groups.
Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the absolute Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for $p$-solvable groups.
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, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
Let $G$ be a finite group. We study the generalized character defined by $Ξ(g)=|G|o(g)$, for $g\in G$, which is closely related to a function that has been very studied recently from a group theoretical point of view.
Let $G$ be a finite solvable group. We prove that if $χ\in{\rm Irr}(G)$ has odd degree and $χ(1)$ is the minimal degree of the non-linear irreducible characters of $G$, then $G/{\rm Ker} χ$ is nilpotent-by-abelian.
Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with 3 irreducible characters is necessarily the cyclic group of order 3. In most cases we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer's height zero conjecture.
Fix a prime $p$ and an integer $n\geq 0$. Among the non-linear irreducible characters of the $p$-groups of order $p^n$, what is the minimum number of elements that take the value 0?
We prove that there exists a universal constant $D$ such that if $p$ is a prime divisor of the index of the Fitting subgroup of a finite group $G$, then the number of conjugacy classes of G is at least $Dp/log_2 p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$.
We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group $G$, with non-linear irreducible characters of $G$ as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by 3. Lastly, the result for $S_n$ is applied to prove the non-equivalence of the metrics on permutations induced from faithful irreducible characters of the group.
We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin-Tits' simple order theorem.