scholarly journals Conjugacy classes and automorphisms of twin groups

2020 ◽  
Vol 32 (5) ◽  
pp. 1095-1108
Author(s):  
Tushar Kanta Naik ◽  
Neha Nanda ◽  
Mahender Singh
Keyword(s):  
Symmetric Group ◽  
Coxeter Group ◽  
Fibonacci Sequence ◽  
Recursive Formula ◽  
Conjugacy Classes ◽  
Structural Aspects

AbstractThe twin group {T_{n}} is a right-angled Coxeter group generated by {n-1} involutions, and the pure twin group {\mathrm{PT}_{n}} is the kernel of the natural surjection from {T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in {T_{n}}, which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in {T_{n}}. We give a new proof of the structure of {\operatorname{Aut}(T_{n})} for {n\geq 3}, and show that {T_{n}} is isomorphic to a subgroup of {\operatorname{Aut}(\mathrm{PT}_{n})} for {n\geq 4}. Finally, we construct a representation of {T_{n}} to {\operatorname{Aut}(F_{n})} for {n\geq 2}.

scholarly journals Conjugacy classes of involutions in Coxeter groups

1982 ◽  
Vol 26 (1) ◽  
pp. 1-15 ◽  
Author(s):  
R.W. Richardson
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Equivalence Classes ◽  
Conjugacy Classes ◽  
Elementary Method

In this paper we give an elementary method for classifying conjugacy classes of involutions in a Coxeter group (W, S). The classification is in terms of (W-equivalence classes of certain subsets of S).

scholarly journals Special involutions and bulky parabolic subgroups in finite Coxeter groups

2005 ◽  
Vol 79 (1) ◽  
pp. 141-147 ◽  
Author(s):  
Götz Pfeiffer ◽  
Gerhard Röhrle
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Hyperplane Arrangement ◽  
Conjugacy Classes ◽  
Parabolic Subgroups ◽  
Finite Coxeter Group

AbstractThe conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

scholarly journals Complete Classification of Degree 7 for Genus 1

2021 ◽  
pp. 594-603
Author(s):  
Peshawa M. Khudhur
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Symmetric Group ◽  
Compact Riemann Surface ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Branched Cover ◽  
Transitive Subgroup ◽  
Genus One

Assume that  is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples (  of permutations in the symmetric group , in which  and s generate a transitive subgroup G of  This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.

The Poset of Conjugacy Classes and Decomposition of Products in the Symmetric Group

1992 ◽  
Vol 35 (2) ◽  
pp. 152-160 ◽  
Author(s):  
François Bédard ◽  
Alain Goupil
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Conjugacy Classes ◽  
The Other ◽  
Linear Combinations ◽  
Graded Poset

AbstractThe action by multiplication of the class of transpositions of the symmetric group on the other conjugacy classes defines a graded poset as described by Birkhoff ([2]). In this paper, the edges of this poset are given a weight and the structure obtained is called the poset of conjugacy classes of the symmetric group. We use weights of chains in the posets to obtain new formulas for the decomposition of products of conjugacy classes of the symmetric group in its group algebra as linear combinations of conjugacy classes and we derive a new identity involving partitions of n.

scholarly journals Irreducible representations of the generalized symmetric group Bmn

1987 ◽  
Vol 29 (1) ◽  
pp. 1-6 ◽  
Author(s):  
M. Saeed-ul-Islam
Keyword(s):  
Symmetric Group ◽  
Wreath Products ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Linear Representations

This paper is devoted to the determining of the irreducible linear representations of the generalized symmetric group (elsewhere written as , Cm ≀ Sn or G(m, 1, n)) by considering the conjugacy classes of and then constructing the same number of inequivalent irreducible linear representations of . These have previously been determined by Kerber [2, Section 5] using Clifford's theory applied to wreath products.

scholarly journals Groups with large conjugacy classes

1977 ◽  
Vol 24 (3) ◽  
pp. 257-265 ◽  
Author(s):  
Bola O. Balogun
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Free Group ◽  
Conjugacy Classes ◽  
A Finite Group

AbstractA finite group is called repetition-free if its conjugacy classes have distinct sizes. It is known that a supersolvable repetition-free group is necessarily isomorphie to Sym(3). the symmetric group on three symbols. Thus the question arises as to whether Sym (3) is the only repetition-free group. In this paper it is proved that if mk denotes the minimum of the orders of the centralizers of elements of a repetition-free group G and mk ≦ 4 then G is isomorphie to Sym (3).

scholarly journals Product of Conjugacy Classes of the Alternating Group An

2012 ◽  
Vol 9 (3) ◽  
pp. 565-568
Author(s):  
Baghdad Science Journal
Keyword(s):  
Positive Integer ◽  
Conjugacy Class ◽  
Symmetric Group ◽  
Nonempty Subset ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Class C

For a nonempty subset X of a group G and a positive integer m , the product of X , denoted by Xm ,is the set Xm = That is , Xm is the subset of G formed by considering all possible ordered products of m elements form X. In the symmetric group Sn, the class Cn (n odd positive integer) split into two conjugacy classes in An denoted Cn+ and Cn- . C+ and C- were used for these two parts of Cn. This work we prove that for some odd n ,the class C of 5- cycle in Sn has the property that = An n 7 and C+ has the property that each element of C+ is conjugate to its inverse, the square of each element of it is the element of C-, these results were used to prove that C+ C- = An exceptional of I (I the identity conjugacy class), when n=5+4k , k>=0.

scholarly journals Complete Quantum Information in the DNA Genetic Code

2020 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Fang Fang ◽  
Klee Irwin
Keyword(s):  
Amino Acids ◽  
Quantum Information ◽  
Symmetric Group ◽  
Genetic Code ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Threefold Symmetry ◽  
Biological Meaning ◽  
Octahedral Group ◽  
Complete Quantum

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature $G=\mathbb{Z}_5 \rtimes H$ where $H=\mathbb{Z}_2 . S_4\cong 2O$ is isomorphic to the binary octahedral group $2O$ and $S_4$ is the symmetric group on four letters/bases. The second group has signature $G=\mathbb{Z}_5 \rtimes GL(2,3)$ and points out a threefold symmetry of base pairings. For those groups, the representations for the $22$ conjugacy classes of $G$ are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the $22$ characters of $G$ attached to those representations are informationally complete. The biological meaning of these coincidences is discussed.

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