ON THE TOTAL COMPONENT OF THE PARTIAL SCHUR MULTIPLIER

2016 ◽  
Vol 100 (3) ◽  
pp. 374-402 ◽  
Author(s):  
H. G. G. DE LIMA ◽  
H. PINEDO
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Closed Field ◽  
Schur Multiplier ◽  
Dihedral Groups ◽  
Infinite Cyclic Group ◽  
Algebraically Closed Field ◽  
Cyclic Groups

In this paper we determine the structure of the total component of the Schur multiplier over an algebraically closed field of some relevant families of groups, such as dihedral groups, dicyclic groups, the infinite cyclic group and the direct product of two finite cyclic groups.

NOETHER SETTINGS FOR CENTRAL EXTENSIONS OF GROUPS WITH ZERO SCHUR MULTIPLIER

2002 ◽  
Vol 01 (01) ◽  
pp. 107-112 ◽  
Author(s):  
ESTHER BENEISH
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cyclic Group ◽  
Central Extension ◽  
Closed Field ◽  
Schur Multiplier ◽  
Central Extensions ◽  
Algebraically Closed Field ◽  
Rational Extension ◽  
A Finite Group

Let G be a finite group, and let F be an algebraically closed field. The rational extension of F, F(G) = F(xg : g ∈ G) with G action given by hxg = xhg for g, h ∈ G, is referred to as the Noether setting for G. We show that if G is a group with zero Schur multiplier, then for any central extension G′ of G, F(G′)G′ and F(G)G are stably equivalent over F. That is, F(G)G is stably rational over F if and only if F(G′)G′ is stably rational over F. In particular, if G is a cyclic group or an abelian group with zero Schur multiplier, then F(G′) is stably rational over F.

scholarly journals The Representations of Quantum Double of Dihedral Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 95-108 ◽  
Author(s):  
Jingcheng Dong ◽  
Huixiang Chen
Keyword(s):  
Group Algebra ◽  
Dihedral Group ◽  
Closed Field ◽  
Quantum Double ◽  
Dihedral Groups ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Finite Dimensional

Let k be an algebraically closed field of odd characteristic p, and let Dn be the dihedral group of order 2n such that p|2n. Let D(kDn) denote the quantum double of the group algebra kDn. In this paper, we describe the structures of all finite-dimensional indecomposable left D(kDn)-modules, equivalently, of all finite-dimensional indecomposable Yetter-Drinfeld kDn-modules, and classify them.

scholarly journals BEING CAYLEY AUTOMATIC IS CLOSED UNDER TAKING WREATH PRODUCT WITH VIRTUALLY CYCLIC GROUPS

2021 ◽  
pp. 1-11
Author(s):  
DMITRY BERDINSKY ◽  
MURRAY ELDER ◽  
JENNIFER TABACK
Keyword(s):  
Computer Science ◽  
Wreath Product ◽  
Cyclic Group ◽  
Wreath Products ◽  
Infinite Cyclic Group ◽  
Cyclic Groups ◽  
Automatic Groups

Abstract We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

scholarly journals Nearly Gorenstein cyclic quotient singularities

2020 ◽  
Author(s):  
Alessio Caminata ◽  
Francesco Strazzanti
Keyword(s):  
Cyclic Group ◽  
Closed Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraically Closed Field ◽  
Quotient Singularities ◽  
Necessary And Sufficient ◽  
Cyclic Quotient Singularities

Abstract We investigate the nearly Gorenstein property among d-dimensional cyclic quotient singularities $$\Bbbk \llbracket x_1,\dots ,x_d\rrbracket ^G$$ k 〚 x 1 , ⋯ , x d 〛 G , where $$\Bbbk $$ k is an algebraically closed field and $$G\subseteq {\text {GL}}(d,\Bbbk )$$ G ⊆ GL ( d , k ) is a finite small cyclic group whose order is invertible in $$\Bbbk $$ k . We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.

scholarly journals A CRITERION FOR THE JACOBSON SEMISIMPLICITY OF THE GREEN RING OF A FINITE TENSOR CATEGORY

2017 ◽  
Vol 60 (1) ◽  
pp. 253-272 ◽  
Author(s):  
ZHIHUA WANG ◽  
LIBIN LI ◽  
YINHUO ZHANG
Keyword(s):  
Cyclic Group ◽  
Jacobson Radical ◽  
Tensor Category ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Green Ring ◽  
Isomorphism Classes

AbstractThis paper deals with the Green ring $\mathcal{G}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$ with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring $\mathcal{G}(\mathcal{C})$, or the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero in K. For the Green ring $\mathcal{G}(\mathcal{C})$ itself, $\mathcal{G}(\mathcal{C})$ is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero. The second part of this paper focuses on the case where $\mathcal{C}=\text{Rep}(\mathbb {k}G)$ for a cyclic group G of order p over a field $\mathbb {k}$ of characteristic p. In this case, the Casimir number of $\mathcal{C}$ is computable and is shown to be 2p2. This leads to a complete description of the Jacobson radical of the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over any field K.

scholarly journals Groups of automorphisms of linearly ordered sets

1976 ◽  
Vol 15 (1) ◽  
pp. 13-32 ◽  
Author(s):  
J.L. Hickman
Keyword(s):  
Simple Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Ordered Set ◽  
Embedding Theorems ◽  
Ordered Sets ◽  
Infinite Cyclic Group ◽  
Groups Of Automorphisms ◽  
Linearly Ordered Set

I show that a group of order-automorphisms of a linearly ordered set can be expressed as an unrestricted direct product in which each factor is either the infinite cyclic group or else a group of order-automorphisms of a densely ordered set. From this a couple of simple group embedding theorems can be derived. The technique used to obtain the main result of this paper was motivated by the Erdös-Hajnal inductive classification of scattered sets.

scholarly journals Oort groups and lifting problems

2008 ◽  
Vol 144 (4) ◽  
pp. 849-866 ◽  
Author(s):  
T. Chinburg ◽  
R. Guralnick ◽  
D. Harbater
Keyword(s):  
Finite Groups ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Cyclic Groups ◽  
Characteristic Two ◽  
Field Of Positive Characteristic

AbstractLet k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

scholarly journals Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

2013 ◽  
Vol 21 (3) ◽  
pp. 207-211
Author(s):  
Hiroshi Yamazaki ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Direct Products ◽  
Cyclic Groups

Summary In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].

scholarly journals Topological rigidity for FJ by the infinite cyclic group

2018 ◽  
Vol 10 (02) ◽  
pp. 421-445
Author(s):  
Kun Wang
Keyword(s):  
Direct Product ◽  
Free Group ◽  
Cyclic Group ◽  
Product Formula ◽  
Addition Formula ◽  
Infinite Cyclic Group ◽  
Torsion Free Group ◽  
Rigidity Results ◽  
Borel Conjecture ◽  
Torsion Free

We call a group FJ if it satisfies the [Formula: see text]- and [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. We show that if [Formula: see text] is FJ, then the simple Borel conjecture (in dimensions [Formula: see text]) holds for every group of the form [Formula: see text]. If in addition [Formula: see text], which is true for all known torsion-free FJ groups, then the bordism Borel conjecture (in dimensions [Formula: see text]) holds for [Formula: see text]. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion-free group [Formula: see text] satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text], then any semi-direct product [Formula: see text] also satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. Our result is indeed more general and implies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in additive categories is closed under extensions of torsion-free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

scholarly journals Isomorphisms of Direct Products of Finite Cyclic Groups

2012 ◽  
Vol 20 (4) ◽  
pp. 343-347
Author(s):  
Kenichi Arai ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Fundamental Theorem ◽  
Chinese Remainder Theorem ◽  
Abelian Groups ◽  
Finite Sequence ◽  
Direct Products ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Relative Prime

Summary In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

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