Classifying finite monomial linear groups of prime degree in characteristic zero

2021 ◽  
pp. 1-39
Author(s):  
Z. Bácskai ◽  
D. L. Flannery ◽  
E. A. O’Brien
Keyword(s):  
Conjugacy Class ◽  
Characteristic Zero ◽  
Structural Information ◽  
Exceptional Case ◽  
Linear Groups ◽  
Complex Field ◽  
Prime Degree ◽  
Generating Sets

Let [Formula: see text] be a prime and let [Formula: see text] be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of [Formula: see text] up to conjugacy. That is, we give a complete and irredundant list of [Formula: see text]-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of [Formula: see text] is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For [Formula: see text], we classify all finite irreducible subgroups of [Formula: see text]. Our classifications are available publicly in Magma.

scholarly journals On a lifting problem of L-packets

2016 ◽  
Vol 152 (9) ◽  
pp. 1800-1850 ◽  
Author(s):  
Bin Xu
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Structural Information ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Symplectic Groups ◽  
Reductive Groups ◽  
Endoscopic Classification

Let $G\subseteq \widetilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and having the same derived group. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of those of $\widetilde{G}$. Motivated by this, we hope to construct the L-packets of $\widetilde{G}$ from those of $G$. The primary example in our mind is when $G=\text{Sp}(2n)$, whose L-packets have been determined by Arthur [The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013)], and $\widetilde{G}=\text{GSp}(2n)$. As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\widetilde{G}$ from those of $G$.

FIXED ELEMENTS OF NONINJECTIVE ENDOMORPHISMS OF POLYNOMIAL ALGEBRAS IN TWO VARIABLES

2016 ◽  
Vol 95 (2) ◽  
pp. 209-213
Author(s):  
YUEYUE LI ◽  
JIE-TAI YU
Keyword(s):  
Associative Algebra ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Free Associative Algebra ◽  
Polynomial Algebras ◽  
Fixed Element ◽  
The Given

Let $A_{2}$ be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of $A_{2}$ and an algorithm to determine whether a given noninjective endomorphism of $A_{2}$ has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism.

Classification of finite groups according to their conjugacy class lengths

2019 ◽  
Vol 22 (1) ◽  
pp. 137-156
Author(s):  
Zeinab Foruzanfar ◽  
İsmai̇l Ş. Güloğlu ◽  
Mehdi Rezaei
Keyword(s):  
Conjugacy Class ◽  
Finite Groups

Abstract In this paper, we classify all finite groups satisfying the following property: their conjugacy class lengths are set-wise relatively prime for any six distinct classes.

scholarly journals Structure insights of the human peroxisomal ABC transporter ALDP

2021 ◽  
Author(s):  
Yutian Jia ◽  
Yanming Zhang ◽  
Jianlin Lei ◽  
Guanghui Yang
Keyword(s):  
Structural Information ◽  
Free State ◽  
Head Group ◽  
Working Mechanism ◽  
Peroxisomal Membrane ◽  
Binding Domains ◽  
Cryo Electron Microscopy ◽  
Ligand Free ◽  
Clinical Description

Adrenoleukodystrophy protein (ALDP) is responsible for the transport of free very-long-chain fatty acids (VLCFAs) and corresponding CoA-esters across the peroxisomal membrane. ALDP belongs to the ATP-binding cassette sub-family D, which is also named as ABCD1. Dysfunction of ALDP leads to peroxisomal metabolic disorder exemplified by X-linked adrenoleukodystrophy (ALD). Hundreds of ALD-causing mutations are identified on ALDP. However, the pathogenic mechanisms of these mutations are restricted to clinical description due to limited structural information. Furthermore, ALDP plays a role in myelin maintenance, which is tightly associated with axon regeneration. Here we report the cryo-electron microscopy (cryo-EM) structure of human ALDP with nominal resolution of 3.4 angstrom in nucleotide free state. The structure of ALDP exhibits a typical assembly of ABC transporters. The nucleotide binding domains (NBDs) displays a ligand free state. ALDP exhibits an inward-open conformation to the cytosol. A short helix is located at the peroxisomal side, which is different from other three members of ABCD transporters. The two transmembrane domains (TMDs) of ALDP form a cavity, in which two lipid-like densities can be recognized as the head group of an coenzyme-A ester of a lipid. This structure provides a framework for understanding the working mechanism of ALDP and classification of the disease-causing mutations.

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

Lifting n-Dimensional Galois Representations

2008 ◽  
Vol 60 (5) ◽  
pp. 1028-1049 ◽  
Author(s):  
Spencer Hamblen
Keyword(s):  
Characteristic Zero ◽  
Galois Representations ◽  
Galois Groups ◽  
Geometric Shape ◽  
General Linear ◽  
Linear Groups ◽  
Selmer Group ◽  
General Linear Groups

AbstractWe investigate the problem of deforming n-dimensional mod p Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of n-dimensional deformations.We then examine under which conditions we may place restrictions on the shape of our deformations at p, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over ℚ of certain infinite subgroups of p-adic general linear groups.

Rational permutation groups containing a full cycle

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
Temha Erkoç ◽  
Utku Yilmaztürk
Keyword(s):  
Symmetric Group ◽  
Proper Subgroup ◽  
Symmetric Groups ◽  
Transitive Permutation Group ◽  
Rational Group ◽  
Prime Degree ◽  
A Finite Group ◽  
Rational Groups ◽  
Complex Characters

AbstractA finite group whose irreducible complex characters are rational valued is called a rational group. Thus, G is a rational group if and only if N G(〈x〉)/C G(〈x〉) ≌ Aut(〈x〉) for every x ∈ G. For example, all symmetric groups and their Sylow 2-subgroups are rational groups. Structure of rational groups have been studied extensively, but the general classification of rational groups has not been able to be done up to now. In this paper, we show that a full symmetric group of prime degree does not have any rational transitive proper subgroup and that a rational doubly transitive permutation group containing a full cycle is the full symmetric group. We also obtain several results related to the study of rational groups.

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