Brauer relations in finite groups

Alex Bartel; Tim Dokchitser

Displaying similar documents to “Brauer relations in finite groups”

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

The unit groups of semisimple group algebras of some non-metabelian groups of order $144$

Gaurav Mittal, Rajendra K. Sharma (2023)

Mathematica Bohemica

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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U (𝔽_{q} G)$ of semisimple group algebra $𝔽_{q} G$ . Here, $q$ denotes the power of a prime, i.e., $q = p^{r}$ for $p$ prime and a positive integer $r$ . Up to isomorphism, there are $6$ groups of order $144$ that have exponent either $36$ or $72$ . Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order $144$ that are a direct product of two...

Twisted group rings of strongly unbounded representation type

Leonid F. Barannyk, Dariusz Klein (2004)

Colloquium Mathematicae

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Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and $S^{λ} G$ a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by $I n d_{m} (S^{λ} G)$ the set of isomorphism classes of indecomposable $S^{λ} G$ -modules of S-rank m. We exhibit rings $S^{λ} G$ for which there exists a function $f_{λ} : ℕ \to ℕ$ such that $f_{λ} (n) \geq n$ and $I n d_{f_{λ} (n)} (S^{λ} G)$ is an infinite set for every natural n > 1. In special cases $f_{λ} (ℕ)$ contains every natural...

Equations in the Hadamard ring of rational functions

Andrea Ferretti, Umberto Zannier (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $K$ be a number field. It is well known that the set of recurrencesequences with entries in $K$ is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume ${a_{n}}$ is a recurrence sequence and suppose that all the $a_{n}$ have a $d^{th}$ root in the field...

(Generalized) filter properties of the amalgamated algebra

Yusof Azimi (2022)

Archivum Mathematicum

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Let $R$ and $S$ be commutative rings with unity, $f : R \to S$ a ring homomorphism and $J$ an ideal of $S$ . Then the subring $R ⋈^{f} J : = {(a, f (a) + j) ∣ a \in R$ and $j \in J}$ of $R \times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$ . In this paper, we determine when $R ⋈^{f} J$ is a (generalized) filter ring.

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $G$ be a group and $m$ be an integer greater than or equal to $2$ . $G$ is said to be $m$ -permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$ , then $G$ is $(1+[z/2])$ -permutable. More bounds are given on the least $m$ such that $G$ is $m$ -permutable.

On weakened $(α, δ)$ -skew Armendariz rings

Alireza Majdabadi Farahani, Mohammad Maghasedi, Farideh Heydari, Hamidagha Tavallaee (2022)

Mathematica Bohemica

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In this note, for a ring endomorphism $α$ and an $α$ -derivation $δ$ of a ring $R$ , the notion of weakened $(α, δ)$ -skew Armendariz rings is introduced as a generalization of $α$ -rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(α, δ)$ -skew Armendariz ring if and only if $T_{n} (R)$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz if and only if $R [x] / (x^{n})$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz ring for any positive integer $n$ .

Obstruction sets and extensions of groups

Francesca Balestrieri (2016)

Acta Arithmetica

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Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X {(_{k})}^{é t, B r} \subset X {(_{k})}^{B r ₁}$ . In the first part, we apply ideas from the proof of $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset \subset_{k}$ are such that $\subset E x t (,_{k})$ , then $X {(_{k})}^{} = X {(_{k})}^{}$ . This allows us to conclude, among other things, that $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ and $X {(_{k})}^{S o l, B r ₁} = X {(_{k})}^{S o l_{k}}$ .

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

Augmentation quotients for Burnside rings of generalized dihedral groups

Shan Chang (2016)

Czechoslovak Mathematical Journal

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Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_{2}$ acting on $H$ by inverting elements, where $C_{2}$ is the cyclic group of order two. Let $Ω (𝒟)$ be the Burnside ring of $𝒟$ , $Δ (𝒟)$ be the augmentation ideal of $Ω (𝒟)$ . Denote by $Δ^{n} (𝒟)$ and $Q_{n} (𝒟)$ the $n$ th power of $Δ (𝒟)$ and the $n$ th consecutive quotient group $Δ^{n} (𝒟) / Δ^{n + 1} (𝒟)$ , respectively. This paper provides an explicit $ℤ$ -basis for $Δ^{n} (𝒟)$ and determines the isomorphism class of $Q_{n} (𝒟)$ for each positive integer $n$ .

Invariants, torsion indices and oriented cohomology of complete flags

Baptiste Calmès, Viktor Petrov, Kirill Zainoulline (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$ . Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$ -theory, Chow groups, Grothendieck’s $K_{0}$ , etc.) with formal group law $F$ . We construct a ring from $F$ and the characters of $T$ , that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁 (G / B)$ where $G / B$ is the variety of Borel subgroups of $G$ . Our main result says that when the...

P-injective group rings

Liang Shen (2020)

Czechoslovak Mathematical Journal

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A ring $R$ is called right P-injective if every homomorphism from a principal right ideal of $R$ to $R_{R}$ can be extended to a homomorphism from $R_{R}$ to $R_{R}$ . Let $R$ be a ring and $G$ a group. Based on a result of Nicholson and Yousif, we prove that the group ring $RG$ is right P-injective if and only if (a) $R$ is right P-injective; (b) $G$ is locally finite; and (c) for any finite subgroup $H$ of $G$ and any principal right ideal $I$ of $RH$ , if $f \in {Hom}_{R} (I_{R}, R_{R})$ , then there exists $g \in {Hom}_{R} ({RH}_{R}, R_{R})$ such that ${g |}_{I} = f$ . Similarly, we also obtain equivalent...

A note on Skolem-Noether algebras

Juncheol Han, Tsiu-Kwen Lee, Sangwon Park (2021)

Czechoslovak Mathematical Journal

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The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let $K$ be a unital commutative ring, not necessarily a field. Given a unital $K$ -algebra $S$ , where $K$ is contained in the center of $S$ , $n \in ℕ$ , the goal of this paper is to study the question: when can a homomorphism $φ : M_{n} (K) \to M_{n} (S)$ be extended to an inner automorphism of $M_{n} (S)$ ? As an application of main results presented in the paper, it is proved that if $S$ is...

Symmetric and reversible properties of bi-amalgamated rings

Antonysamy Aruldoss, Chelliah Selvaraj (2024)

Czechoslovak Mathematical Journal

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Let $f : A \to B$ and $g : A \to C$ be two ring homomorphisms and let $K$ and $K^{'}$ be two ideals of $B$ and $C$ , respectively, such that $f^{- 1} (K) = g^{- 1} (K^{'})$ . We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A ⋈^{f, g} (K, K^{'})$ of $A$ with $(B, C)$ along $(K, K^{'})$ with respect to $(f, g)$ .