scholarly journals Canonical Forms for Dihedral and Symmetric Groups

10.37236/8583 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Robert Shwartz
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Canonical Forms ◽  
Descent Set ◽  
Descent Set Statistics

In this paper we introduce the elementary factorization of the standard OGS for the symmetric group, and show how it encodes the inversion and descent set statistics. Proofs follow from exchange laws for powers of Coxeter elements in the principal flag.

scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

scholarly journals The multistep homology of the simplex and representations of symmetric groups

2019 ◽  
Vol 169 (2) ◽  
pp. 231-253
Author(s):  
MARK WILDON
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Explicit Construction ◽  
Fixed Size ◽  
Simplicial Homology ◽  
Chain Complexes ◽  
Basic Spin ◽  
Characteristic Two

AbstractThe symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

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.

scholarly journals Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1178
Author(s):  
Athirah Nawawi ◽  
Sharifah Kartini Said Husain ◽  
Muhammad Rezal Kamel Ariffin
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Commutativity Property

A commuting graph is a graph denoted by C ( G , X ) where G is any group and X, a subset of a group G, is a set of vertices for C ( G , X ) . Two distinct vertices, x , y ∈ X , will be connected by an edge if the commutativity property is satisfied or x y = y x . This study presents results for the connectivity of C ( G , X ) when G is a symmetric group of degree n, Sym ( n ) , and X is a conjugacy class of elements of order three in G.

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

scholarly journals On alternating and symmetric groups which are quasi OD-characterizable

2017 ◽  
Vol 16 (04) ◽  
pp. 1750065 ◽  
Author(s):  
Ali Reza Moghaddamfar
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Symmetric Groups ◽  
Degree Pattern ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

t-Representability of maximal subgroups of symmetric groups

2018 ◽  
Vol 9 (1) ◽  
pp. 27
Author(s):  
Sini P
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Maximal Subgroups ◽  
Group Of Homeomorphisms

A subgroup \(H\) of the group \(S(X)\) of all permutations of a set \(X\) is called \(t\)−representable on \(X\) if there exists a topology \(T\) on \(X\) such that the group of homeomorphisms of \((X, T ) = K\). In this paper we study the \(t\)-representability of maximal subgroups of the symmetric group.

scholarly journals On Solutions of xd = 1 In Symmetric Groups

1955 ◽  
Vol 7 ◽  
pp. 159-168 ◽  
Author(s):  
Leo Moser
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Number Of Solutions

1. Introduction. Several recent papers have dealt with the number of solutions of xd = 1 in Sn, the symmetric group of degree n. Let us denote this number by An,d and let An,2 = Tn.

COUNTING ELEMENTS IN THE SYMMETRIC GROUP

2009 ◽  
Vol 19 (03) ◽  
pp. 305-313 ◽  
Author(s):  
DAVID EL-CHAI BEN-EZRA
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Combinatorial Identities ◽  
Subgroup Growth

By using simple ideas from subgroup growth of pro-finite groups we deduce some combinatorial identities on generating functions counting various elements in symmetric groups.

THE ESSENTIAL RANK OF FUSION SYSTEMS OF BLOCKS OF SYMMETRIC GROUPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250002 ◽  
Author(s):  
JIANBEI AN ◽  
HEIKO DIETRICH
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Fusion System ◽  
Fusion Systems ◽  
Local Subgroup

Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, bD) be a Sylow B-subgroup of S(n). We consider the fusion system [Formula: see text] and determine a precise formula for its essential rank. In addition, the p-blocks B which admit a p-local subgroup of S(n) controlling B-fusion are characterized.

Sign in / Sign up

Export Citation Format

Share Document