scholarly journals Q-conjugacy character table for the non-rigid group of 2,3-dimethylbutane

2009 ◽  
Vol 74 (1) ◽  
pp. 45-52 ◽  
Author(s):  
Reza Darafsheh ◽  
Ali Moghani
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Direct Product ◽  
Cyclic Group ◽  
Conjugacy Classes ◽  
Character Table ◽  
Irreducible Characters ◽  
Dominant Classes ◽  
A Finite Group ◽  
Rigid Group

Maturated and unmaturated groups were introduced by the Japanese chemist Shinsaku Fujita, who used them in the markaracter table and the Q-conjugacy character table of a finite group. He then applied his results in this area of research to enumerate isomers of molecules. Using the non-rigid group theory, it was shown by the second author that the full non-rigid (f-NRG) group of 2,3- -dimethylbutane is isomorphic to the group (Z3?Z3?Z3?Z3):Z2 of order 162 with 54 conjugacy classes. Here (Z3?Z3?Z3?Z3):Z2 denotes the semi direct product of four copies of Z3 by Z2, where Zn is a cyclic group of order n. In this paper, it is shown with the GAP program that this group has 30 dominant classes (similarly, Q-conjugacy characters) and that 24 of them are unmatured (similarly, Q-conjugacy characters such that they are the sum of two irreducible characters). Then, the Q-conjugacy character table of the unmatured full non-rigid group 2,3-dimethylbutane is derived.

scholarly journals Informationally Complete Characters for Quark and Lepton Mixings

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1000 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Hilbert Space ◽  
Finite Group ◽  
Three Dimensional ◽  
Conjugacy Classes ◽  
Character Table ◽  
Three Generations ◽  
Dimensional Hilbert Space ◽  
Mixing Patterns ◽  
A Finite Group

A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters κ of a finite group G. Here, we introduce a d-dimensional Hilbert space with d = c c ( G ) , the number of conjugacy classes of G. Groups under consideration should follow two rules, (a) the character table contains both two- and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measurements under the action of a d-dimensional Pauli group over the characters of these representations. Groups with small d that satisfy these rules coincide in a large part with viable ones derived so far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing matrices.

scholarly journals The Number of Conjugacy Classes and Irreducible Characters in a Finite Group

2019 ◽  
pp. 181-190
Author(s):  
Remigius Okeke Aja ◽  
Uchenna Emmanuel Obasi ◽  
Everestus Obinwanne Eze
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
Disjoint Cycles ◽  
A Finite Group

In this paper, number of conjugacy classes and irreducible characters in a non-abelian group of order $2^6$ are investigated using cycle pattern of elements. Through the exploits of commutator and representation of elements as a product of disjoint cycles, the number of conjugacy classes is obtained which extends some results in literature.

scholarly journals An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups

1987 ◽  
Vol 30 (1) ◽  
pp. 7-22 ◽  
Author(s):  
Zvi Arad ◽  
Elsa Fisman
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
Basic Concepts ◽  
A Finite Group

It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G. The purpose of this article is to give some analogous properties between these basic concepts.

Powers of irreducible characters and conjugacy classes in finite groups

2014 ◽  
Vol 13 (08) ◽  
pp. 1450067 ◽  
Author(s):  
M. R. Darafsheh ◽  
S. M. Robati
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Upper Bounds ◽  
Conjugacy Classes ◽  
Covering Number ◽  
Irreducible Characters ◽  
A Finite Group

Let G be a finite group. We define the derived covering number and the derived character covering number of G, denoted respectively by dcn (G) and dccn (G), as the smallest positive integer n such that Cn = G′ for all non-central conjugacy classes C of G and Irr ((χn)G′) = Irr (G′) for all nonlinear irreducible characters χ of G, respectively. In this paper, we obtain some results on dcn and dccn for a finite group G, such as the existence of these numbers and upper bounds on them.

scholarly journals A submatrix of the character table

2000 ◽  
Vol 62 (3) ◽  
pp. 525-528
Author(s):  
Gabriel Navarro
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Conjugacy Classes ◽  
Character Table ◽  
Maximum Rank ◽  
A Finite Group

Let G be a finite group and let p be a prime number. We consider the Submatrix of the character table of G whose rows are indexed by the characters in blocks of maximal defect, and whose columns are indexed by the conjugacy classes of P′-size. We prove that this matrix has maximum rank.

scholarly journals Orbit sizes, local subgroups and chains of p-groups

2001 ◽  
Vol 71 (3) ◽  
pp. 325-338 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Group ◽  
General Theory ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
A Finite Group

AbstractLet G be a finite group that acts on a finite group V, and let p be a prime that does not divide the order of V. Then the p-parts of the orbit sizes are the same in the actions of G on the sets of conjugacy classes and irreducible characters of V. This result is derived as a consequence of some general theory relating orbits and chains of p-subgroups of a group.

scholarly journals Towards the classification of finite simple groups with exactly three or four supercharacter theories

2018 ◽  
Vol 11 (05) ◽  
pp. 1850096 ◽  
Author(s):  
A. R. Ashrafi ◽  
F. Koorepazan-Moftakhar
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Compatibility Conditions ◽  
Irreducible Characters ◽  
Supercharacter Theory ◽  
A Finite Group

A supercharacter theory for a finite group [Formula: see text] is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions. The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.

scholarly journals Computing characters of groups with central subgroups

2013 ◽  
Vol 16 ◽  
pp. 398-406
Author(s):  
Vahid Dabbaghian ◽  
John D. Dixon
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
Linear Character ◽  
Character Tables ◽  
A Finite Group

AbstractThe so-called Burnside–Dixon–Schneider (BDS) method, currently used as the default method of computing character tables in GAP for groups which are not solvable, is often inefficient in dealing with groups with large centres. If $G$ is a finite group with centre $Z$ and $\lambda $ a linear character of $Z$, then we describe a method of computing the set $\mathrm{Irr} (G, \lambda )$ of irreducible characters $\chi $ of $G$ whose restriction ${\chi }_{Z} $ is a multiple of $\lambda $. This modification of the BDS method involves only $\vert \mathrm{Irr} (G, \lambda )\vert $ conjugacy classes of $G$ and so is relatively fast. A generalization of the method can be applied to computation of small sets of characters of groups with a solvable normal subgroup.Supplementary materials are available with this article.

THE CONJUGACY CLASSES OF A SUBGROUP $S_{n}^{m} : C_{m}$ OF Smn, prime m

2008 ◽  
Vol 18 (04) ◽  
pp. 705-717
Author(s):  
KENNETH ZIMBA ◽  
MERIAM RABOSHAKGA
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Direct Product ◽  
Cyclic Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Character Table ◽  
Split Extension ◽  
Alternative Method ◽  
Positive Integers

The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let n,m be positive integers and [Formula: see text] be the direct product of m copies of the symmetric group Sn of degree n. Then [Formula: see text] is a subgroup of the symmetric group Smn of degree m × n. Let g∈Smn, of type [mn] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of Sn act. Then g permutes the elements of the copies of Sn in [Formula: see text] and generates a cyclic group Cm = 〈g〉 of order m. The wreath product of Sn with Cm is a split extension or semi-direct product of [Formula: see text] by Cm, denoted by [Formula: see text]. It is clear that [Formula: see text] is a subgroup of the symmetric group Smn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of [Formula: see text], where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group [Formula: see text], this method is useful in the construction of Fischer–Clifford matrices of the group [Formula: see text]. These Fischer–Clifford matrices are useful in the construction of the character table of [Formula: see text].

scholarly journals Parameterization of Irreducible Characters for p-Solvable Groups

2012 ◽  
Vol 19 (01) ◽  
pp. 1-40 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
Outer Automorphisms ◽  
A Finite Group

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

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