Unitary Representations of Generalized Symmetric Groups

1969 ◽  
Vol 21 ◽  
pp. 28-38 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Symmetric Groups ◽  
Unitary Representations ◽  
Complex Field ◽  
Permutation Matrices ◽  
Image Position

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

scholarly journals Note on a paper of Tsuzuku

1964 ◽  
Vol 6 (4) ◽  
pp. 196-197
Author(s):  
H. K. Farahat
Keyword(s):  
Symmetric Group ◽  
Commutative Ring ◽  
Permutation Group ◽  
Matrix Representation ◽  
Invertible Element ◽  
Unit Element ◽  
Transitive Permutation Group ◽  
Natural Representation ◽  
Correct Proof ◽  
Permutation Matrices

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

scholarly journals Finitely presented algebras defined by permutation relations of dihedral type

2016 ◽  
Vol 26 (01) ◽  
pp. 171-202 ◽  
Author(s):  
Ferran Cedó ◽  
Eric Jespers ◽  
Georg Klein
Keyword(s):  
Normal Subgroup ◽  
Normal Form ◽  
Symmetric Group ◽  
Cyclic Group ◽  
Index Formula ◽  
Universal Group ◽  
Product Group ◽  
Unique Product ◽  
Dihedral Type ◽  
Finitely Presented

The class of finitely presented algebras over a field [Formula: see text] with a set of generators [Formula: see text] and defined by homogeneous relations of the form [Formula: see text], where [Formula: see text] runs through a subset [Formula: see text] of the symmetric group [Formula: see text] of degree [Formula: see text], is investigated. Groups [Formula: see text] in which the cyclic group [Formula: see text] is a normal subgroup of index [Formula: see text] are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid [Formula: see text] with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group [Formula: see text] of [Formula: see text] is a unique product group, and it is the central localization of a cancellative subsemigroup of [Formula: see text]. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra [Formula: see text] is semiprimitive.

scholarly journals On induced permutation matrices and the symmetric group

1936 ◽  
Vol 5 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Symmetric Group ◽  
Matrix Representation ◽  
Zero Element ◽  
Natural Order ◽  
Third Order ◽  
The Third ◽  
Permutation Matrices ◽  
The Matrix ◽  
Matrix Relation ◽  
Image Position

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third orderare permutation matrices. It is convenient to denote them bywhere the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

scholarly journals Two Notes on Matrices

1962 ◽  
Vol 5 (3) ◽  
pp. 109-113 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Cyclic Group ◽  
Matrix Representation ◽  
Circulant Matrix ◽  
Diagonal Form ◽  
Complex Field ◽  
Group Matrix ◽  
Matrix Coefficients ◽  
Image Position ◽  
Latent Roots

1. The properties of the circulant determinant or the circulant matrix are familiar. The circulant matrix C of order 4 x 4, with elements in the complex field, will serve for illustration.The four matrix coefficients of c0, c1 c2, c3 form a reducible matrix representation of the cyclic group ℐ4, so that C is a group matrix for this. Let ω be a primitive 4th root of 1. Then Ω as below, its columns being normalized latent vectors of C,is unitary and symmetric, and reduces Cto diagonal form thus,where the μr, the latent roots of C, are given byAll of the above extends naturally to the n x n case.

scholarly journals A Novel Technique for the Construction of Safe Substitution Boxes Based on Cyclic and Symmetric Groups

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Abdul Razaq ◽  
Hanan A. Al-Olayan ◽  
Atta Ullah ◽  
Arshad Riaz ◽  
Adil Waheed
Keyword(s):  
Symmetric Group ◽  
Cyclic Group ◽  
Symmetric Groups ◽  
Galois Field ◽  
Substitution Box ◽  
Novel Technique ◽  
Initial Sequence ◽  
Substitution Boxes ◽  
The Matrix ◽  
Better Than

In the literature, different algebraic techniques have been applied on Galois field GF(28) to construct substitution boxes. In this paper, instead of Galois field GF(28), we use a cyclic group C255 in the formation of proposed substitution box. The construction proposed S-box involves three simple steps. In the first step, we introduce a special type of transformation T of order 255 to generate C255. Next, we adjoin 0 to C255 and write the elements of C255∪0 in 16×16 matrix to destroy the initial sequence 0,1,2,…,255. In the 2nd step, the randomness in the data is increased by applying certain permutations of the symmetric group S16 on rows and columns of the matrix. In the last step we consider the symmetric group S256, and positions of the elements of the matrix obtained in step 2 are changed by its certain permutations to construct the suggested S-box. The strength of our S-box to work against cryptanalysis is checked through various tests. The results are then compared with the famous S-boxes. The comparison shows that the ability of our S-box to create confusion is better than most of the famous S-boxes.

scholarly journals On the natural representation of the symmetric groups

1962 ◽  
Vol 5 (3) ◽  
pp. 121-136 ◽  
Author(s):  
H. K. Farahat
Keyword(s):  
Finite Number ◽  
Symmetric Group ◽  
Commutative Ring ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Unit Element ◽  
Natural Representation ◽  
Image Position ◽  
Obvious Manner

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ,

scholarly journals Group closures of one-to-one transformations

2001 ◽  
Vol 64 (2) ◽  
pp. 177-188 ◽  
Author(s):  
Inessa Levi
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternating Group ◽  
One To One ◽  
Necessary And Sufficient ◽  
Infinite Set

For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

scholarly journals On Multiple Transitivity of Permutation Groups

1961 ◽  
Vol 18 ◽  
pp. 93-109 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Irreducible Character ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Irreducible Characters ◽  
Interesting Theorem

It is well known that a doubly transitive group has an irreducible character χ1 such that χ1(R) = α(R) − 1 for any element R of and a quadruply transitive group has irreducible characters χ3 and χ3 such that χ2(R) = where α(R) and β(R) are respectively the numbers of one cycles and two cycles contained in R. G. Frobenius was led to this fact in the connection with characters of the symmetric groups and he proved the following interesting theorem: if a permutation group of degree n is t-ply transitive, then any irreducible character of the symmetric group of degree n with dimension at most equal to is an irreducible character of .

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