scholarly journals Approximation by representative functions on the complete product of S3

2008 ◽  
Vol 21 (3) ◽  
pp. 327-337 ◽  
Author(s):  
Rodolfo Toledo
Keyword(s):  
Fourier Analysis ◽  
Symmetric Group ◽  
Finite Groups ◽  
Concrete Case ◽  
Complete Product

This work summarizes some statements with respect to Fourier analysis on the complete product of not necessarily commutative finite groups, achieved recently. In particular we devotes attention to a concrete case: the complete product of the symmetric group on 3 elements. The aim of this work is to emphasize the differences between this noncommutative structure and the commutative cases. .

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.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

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.

scholarly journals Spectrum of commuting graphs of some classes of finite groups

MATEMATIKA ◽  
2017 ◽  
Vol 33 (1) ◽  
pp. 87 ◽  
Author(s):  
Rajat Kanti Nath ◽  
Jutirekha Dutta
Keyword(s):  
Abelian Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Commuting Graph

In this paper, we initiate the study of spectrum of the commuting graphs of finite non-abelian groups. We first compute the spectrum of this graph for several classes of finite groups, in particular AC-groups. We show that the commuting graphs of finite non-abelian AC-groups are integral. We also show that the commuting graph of a finite non-abelian group G is integral if G is not isomorphic to the symmetric group of degree 4 and the commuting graph of G is planar. Further, it is shown that the commuting graph of G is integral if its commuting graph is toroidal.

scholarly journals Fourier Analysis on Finite Groups and the Lovász ϑ-Number of Cayley Graphs

2014 ◽  
Vol 23 (2) ◽  
pp. 146-152 ◽  
Author(s):  
Evan DeCorte ◽  
David de Laat ◽  
Frank Vallentin
Keyword(s):  
Fourier Analysis ◽  
Finite Groups ◽  
Cayley Graphs

scholarly journals Rational valued and real valued projective characters of finite groups

1980 ◽  
Vol 21 (1) ◽  
pp. 23-28 ◽  
Author(s):  
J. F. Humphreys
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
General Information ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Character ◽  
Definition Of ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Spin Characters

It is well-known [3; V.13.7] that each irreducible complex character of a finite group G is rational valued if and only if for each integer m coprime to the order of G and each g ∈ G, g is conjugate to gm. In particular, for each positive integer n, the symmetric group on n symbols, S(n), has all its irreducible characters rational valued. The situation for projective characters is quite different. In [5], Morris gives tables of the spin characters of S(n) for n ≤ 13 as well as general information about the values of these characters for any symmetric group. It can be seen from these results that in no case are all the spin characters of S(n) rational valued and, indeed, for n ≥ 6 these characters are not even all real valued. In section 2 of this note, we obtain a necessary and sufficient condition for each irreducible character of a group G associated with a 2-cocycle α to be rational valued. A corresponding result for real valued projective characters is discussed in section 3. Section 1 contains preliminary definitions and notation, including the definition of projective characters given in [2].

scholarly journals Character theory of infinite wreath products

2005 ◽  
Vol 2005 (9) ◽  
pp. 1365-1379 ◽  
Author(s):  
Robert Boyer
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Finite Groups ◽  
Wreath Products ◽  
Character Theory ◽  
Group Algebras ◽  
Inductive Limits ◽  
Infinite Symmetric Group ◽  
The Relationship ◽  
Asymptotic Character

The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” TheK0-invariant of the groupC∗-algebra is also determined.

Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design

2005 ◽  
Author(s):  
Radomir S. Stanković ◽  
Claudio Moraga ◽  
Jaakko T. Astola
Keyword(s):  
Signal Processing ◽  
Fourier Analysis ◽  
System Design ◽  
Finite Groups
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