Explanipedia

Character Values of $p$-solvable groups on picky elements Open

Alexander Moretó, Gabriel Navarro, Noelia Rizo · 2025

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.

Decomposition numbers in the principal block and Sylow normalisers Open

Gunter Malle, Noelia Rizo · 2025

Brauer’s problem 21 for principal blocks Open

Alexander Moretó, Noelia Rizo, A. A. Schaeffer Fry · 2025

Problem 21 of Brauer’s list of problems from 1963 asks whether for any positive integer there are finitely many isomorphism classes of groups that occur as the defect group of a block with irreducible characters. We solve this problem fo…

Characters and Sylow subgroup abelianization Open

Eugenio Giannelli, Noelia Rizo, A. A. Schaeffer Fry, Carolina Vallejo · 2025

Group theory

Fixed point ratios, Sylow numbers and coverings of $p$-elements in finite groups Open

Robert M. Guralnick, Attila Maróti, Juan Martínez Madrid, Alexander Moretó, Noelia Rizo · 2024

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 …

Characters and Sylow $3$-subgroup abelianization Open

Eugenio Giannelli, Noelia Rizo, A. A. Schaeffer Fry, Carolina Vallejo · 2024

We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 3-block of G, settling a conjecture put forward by Navarro, Sambale…

Decomposition numbers in the principal block and Sylow normalisers Open

Gunter Malle, Noelia Rizo · 2024

If G is a finite group and p is a prime number, we investigate the relationship between the p-modular decomposition numbers of characters of height zero in the principal p-block of G and the p-local structure of G.

Minimal heights and defect groups with two character degrees Open

Gunter Malle, Alexander Moretó, Noelia Rizo · 2024

Conjecture A of [3] 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 b…

A Brauer--Galois height zero conjecture Open

Gunter Malle, Alexander Moretó, Noelia Rizo, A. A. Schaeffer Fry · 2024

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 c…

The Alperin Weight Conjecture and the Glauberman correspondence via character triples Open

Jorge Martínez, Noelia Rizo, Damiano Rossi · 2023

Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes …

Brauer's problem 21 for principal blocks Open

Alexander Moretó, Noelia Rizo, A. A. Schaeffer Fry · 2023

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 …

The blocks with five irreducible characters Open

J. Miquel Martínez, Noelia Rizo, Lucía Sanus · 2023

Let $G$ be a finite group, $p$ a prime and $B$ a Brauer $p$-block of $G$ with defect group $D$. We prove that if the number of irreducible ordinary characters in $B$ is $5$ then $D\cong C_5, C_7, D_8$ or $Q_8$, assuming that the Alperin--M…

Minimal heights and defect groups with two character degrees Open

Gunter Malle, Alexander Moretó, Noelia Rizo · 2023

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…

Principal Blocks for Different Primes, II Open

Gabriel Navarro, Noelia Rizo, A. A. Schaeffer Fry · 2022

If G is a finite group, we have proposed three new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper,we give strong support to the validity of Conjectures B and C.

The blocks with four irreducible characters Open

Jorge Martínez, Noelia Rizo, Lucía Sanus · 2022

Suppose that is a Brauer ‐block of a finite group with defect group . If exactly contains four ordinary irreducible characters, then we show that has order four or five, assuming the Alperin–McKay conjecture holds for .

Principal blocks for different primes, I Open

Gabriel Navarro, Noelia Rizo, A. A. Schaeffer Fry · 2022

We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.

Principal blocks for different primes, II Open

Gabriel Navarro, Noelia Rizo, A. A. Schaeffer Fry · 2022

If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.

Principal blocks for different primes, I Open

Gabriel Navarro, Noelia Rizo, A. A. Schaeffer Fry · 2022

We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.

Brauer character degrees and Sylow normalizers Open

Lorenzo Bonazzi, Gabriel Navarro, Noelia Rizo, Lucía Sanus · 2022

If p and q are primes, and G is a p -solvable finite group, it is possible to detect that a q -Sylow normalizer is contained in a p -Sylow normalizer using the character table of G . This is characterized in terms of the degrees of p -Brau…

The blocks with four irreducible characters Open

J. Miquel Martínez, Noelia Rizo, Lucía Sanus · 2021

Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.

Principal blocks with 5 irreducible characters Open

Noelia Rizo, A. A. Schaeffer Fry, Carolina Vallejo · 2021

Kernels of p'-degree irreducible characters Open

Alexander Moretó, Noelia Rizo · 2021

We prove a p'-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on p-parts of character codegrees.

Characters and generation of Sylow 2-subgroups Open

Gabriel Navarro, Noelia Rizo, A. A. Schaeffer Fry, Carolina Vallejo · 2021

We show that the character table of a finite group G determines whether a Sylow 2-subgroup of G is generated by 2 elements, in terms of the Galois action on characters. Our proof of this result requires the use of the Classification of Fin…

Principal blocks with 5 irreducible characters Open

Noelia Rizo, A. A. Schaeffer Fry, Carolina Vallejo · 2021

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

Galois action on the principal block and cyclic Sylow subgroups Open

Noelia Rizo, A. A. Schaeffer Fry, Carolina Vallejo · 2020

We characterize finite groups [math] having a cyclic Sylow [math] -subgroup in terms of the action of a specific Galois automorphism on the principal [math] -block of [math] , for [math] . We show that the analog statement for blocks with …

Groups with few 𝑝’-character degrees in the principal block Open

Eugenio Giannelli, Noelia Rizo, Benjamin Sambale, A. A. Schaeffer Fry · 2020

Let be a prime and let be a finite group. We prove that is -solvable of -length at most if there are at most two distinct -character degrees in the principal -block of . This generalizes a theorem of Isaacs–Smith as well as a recent re…

Groups with few $p'$-character degrees in the principal block Open

Eugenio Giannelli, Noelia Rizo, Benjamin Sambale, A. A. Schaeffer Fry · 2020

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 the…

Groups with few p′-character degrees Open

Eugenio Giannelli, Noelia Rizo, Mandi Schaeffer Fry · 2020

Unitary $t$-groups Open

Eiichi Bannaĭ, Gabriel Navarro, Noelia Rizo, Pham Huu Tiep · 2019

Relying on the main results of [GT], we classify all unitary $t$-groups for $t \\geq 2$ in any dimension $d \\geq 2$. We also show that there is essentially a unique unitary 4-group, which is also a unitary 5-group, but not a unitary $t$-g…

Unitary t-groups Open

Eiichi Bannaĭ, Gabriel Navarro, Noelia Rizo, Pham Huu Tiep · 2018

Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary $4$-group, which is also a unitary $5$-group, but not a…

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