scholarly journals A Constructive Method for Generating Short Presentations for the Symmetric Groups Sm+n, S2m and Smn

2021 ◽  
pp. 37-47
Author(s):  
D. Samaila ◽  
G. N. Shu’aibu ◽  
B. A. Modu
Keyword(s):  
Finite Groups ◽  
Cartesian Product ◽  
Symmetric Groups ◽  
Short Length ◽  
Constructive Method ◽  
Standing Problem ◽  
Product Of Groups

A long-standing problem is how to create a short-length presentation for finite groups of degree n. This paper aimed at presenting a concrete method for generating presentations for the groups Sm+n, S2m and Smn for all m,nÎZ+ with fewer relations than the existing literature from the presentations of Sm and Sn. The aim is achieved by considering finite groups acting on sets and Cartesian product of groups which lead to the construction of multiple transformations as representatives of some finite groups.

The Use of S-Functions in Combinatorial Analysis

1968 ◽  
Vol 20 ◽  
pp. 808-841 ◽  
Author(s):  
Ronald C. Read
Keyword(s):  
Graph Theory ◽  
Finite Groups ◽  
Symmetric Functions ◽  
Symmetric Groups ◽  
Combinatorial Problems ◽  
Combinatorial Analysis

The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.

Suatu Hubungan Kumpulan Pusat–2 dengan Teori Kebarangkalian

2012 ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Hasimah Sapiri
Keyword(s):  
Probability Theory ◽  
Finite Group ◽  
Finite Groups ◽  
Random Element ◽  
Symmetric Groups ◽  
Finite Rings ◽  
Nonabelian Group ◽  
Central Group ◽  
Upper Limit

Penentuan darjah keabelanan bagi suatu kumpulan tak abelan telah diperkenalkan untuk kumpulan simetri oleh Erdos dan Turan [1]. Dalam tahun 1973, Gustafson [2] mengkajinya bagi kumpulan terhingga sementara MacHale [3] mengkajinya bagi gelanggang terhingga dalam tahun 1976. Dalam kajian ini, beberapa keputusan yang berkaitan dengan Pn(G), kebarangkalian bahawa suatu unsur rawak dengan kuasa ke–n dalam suatu kumpulan pusat–2 G adalah kalis tukar tertib dengan unsur rawak yang lain dalam kumpulan yang sama, akan diberikan. Seterusnya, batas atas bagi P2(G) diperoleh. Kata kunci: Teori kebarangkalian, teori kumpulan, kumpulan terhingga, kalis tukar tertib The determination of the abelianness of a nonabelian group has been introduced for symmetric groups by Erdos & Turan [1]. In 1973, Gustafson [2] did this research for the finite groups while MacHale [3] determined the abelianness for finite rings in 1976. In this research, some results on Pn(G), the probability that the n–th power of a random element in a 2–central group G commutes with another random element from the same group, will be presented. Furthermore, the upper limit of P2(G) is obtained. Key words: Probability theory, group theory, finite group, commutative

Strong Gelfand pairs of symmetric groups

2020 ◽  
pp. 2150054
Author(s):  
Gradin Anderson ◽  
Stephen P. Humphries ◽  
Nathan Nicholson
Keyword(s):  
Commutative Ring ◽  
Finite Groups ◽  
Symmetric Groups ◽  
Maximal Dimension ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Schur Ring

A strong Gelfand pair is a pair [Formula: see text], of finite groups such that the Schur ring determined by the [Formula: see text]-classes [Formula: see text], is a commutative ring. We find all strong Gelfand pairs [Formula: see text]. We also define an extra strong Gelfand pair [Formula: see text], this being a strong Gelfand pair of maximal dimension, and show that in this case [Formula: see text] must be abelian.

scholarly journals CRITICAL PERCOLATION OF FREE PRODUCT OF GROUPS

2008 ◽  
Vol 18 (04) ◽  
pp. 683-704 ◽  
Author(s):  
IVA KOZÁKOVÁ
Keyword(s):  
Cayley Graph ◽  
Free Product ◽  
Finite Groups ◽  
Cluster Size ◽  
Critical Probability ◽  
Critical Percolation ◽  
Residually Finite Groups ◽  
Quotient Groups ◽  
Free Product Of Groups ◽  
Product Of Groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability pc of a free product G1 * G2 * ⋯ * Gn of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups G1, G2, …, Gn. For finite groups this equation is polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that pc for the Cayley graph of the modular group PSL2(ℤ) (with the standard generators) is 0.5199…, the unique root of the polynomial 2p5 - 6p4 + 2p3 + 4p2 - 1 in the interval (0, 1). In the case when groups Gi can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of G1 * G2 * ⋯ * Gn and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, p exp of the free product is just the minimum of p exp for the factors.

Orthogonal $*$-basis of relative symmetry classes of polynomials

2015 ◽  
Vol 30 ◽  
Author(s):  
Kijti Rodtes
Keyword(s):  
Finite Groups ◽  
Symmetric Groups ◽  
Orthogonal Basis ◽  
Irreducible Characters ◽  
Symmetry Classes

In this note, the existence of orthogonal ∗-basis of the symmetry classes of polynomials is discussed. Analogously to the orthogonal ∗-basis of symmetry classes of tensor, some criteria for the existence of the basis for finite groups are provided. A condition for the existence of such basis of symmetry classes of polynomials associated to symmetric groups and some irreducible characters is also investigated.

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.

scholarly journals Young modules for symmetric groups

2001 ◽  
Vol 71 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Karin Erdmann
Keyword(s):  
Representation Theory ◽  
Finite Groups ◽  
Symmetric Groups ◽  
General Linear ◽  
Linear Groups ◽  
Schur Algebras ◽  
General Linear Groups ◽  
Green Correspondence

AbstractLet K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

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