scholarly journals Computing the conjugacy classes and character table of a non-split extension 26·(25:S6) from a split extension 26:(25:S6)

2020 ◽  
Vol 5 (3) ◽  
pp. 2113-2125
Author(s):  
Abraham Love Prins ◽  
Keyword(s):  
Conjugacy Classes ◽  
Character Table ◽  
Split Extension

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].

The Fischer–Clifford Matrices of a Maximal Subgroup of the Sporadic Simple Group of Held

2007 ◽  
Vol 14 (01) ◽  
pp. 135-142 ◽  
Author(s):  
Faryad Ali
Keyword(s):  
Simple Group ◽  
Computer Algebra ◽  
Local Structure ◽  
Computer Algebra System ◽  
Conjugacy Classes ◽  
Character Table ◽  
Maximal Subgroups ◽  
Split Extension ◽  
Sporadic Simple Group ◽  
Held Group

The Held group He discovered by Held [10] is a sporadic simple group of order 4030387200 = 210.33.52.73.17. The group He has 11 conjugacy classes of maximal subgroups as determined by Butler [5] and listed in the 𝔸𝕋𝕃𝔸𝕊. Held himself determined much of the local structure of He as well as the conjugacy classes of its elements. Thompson calculated the character table of He . In the present paper, we determine the Fischer–Clifford matrices and hence compute the character table of the non-split extension 3·S7, which is a maximal subgroups of He of index 226560 using the technique of Fischer–Clifford matrices. Most of the computations were carried out with the aid of the computer algebra system 𝔾𝔸ℙ.

A NOTE ON THE NUMBER OF ZEROS IN THE COLUMNS OF THE CHARACTER TABLE OF A GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250150 ◽  
Author(s):  
JINSHAN ZHANG ◽  
ZHENCAI SHEN ◽  
SHULIN WU
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Table ◽  
Number Of Zeros

The finite groups in which every irreducible character vanishes on at most three conjugacy classes were characterized [J. Group Theory13 (2010) 799–819]. Dually, we investigate the finite groups whose columns contain a small number of zeros in the character table.

The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of Fi24

2010 ◽  
Vol 17 (03) ◽  
pp. 389-414 ◽  
Author(s):  
Faryad Ali ◽  
Jamshid Moori
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Maximal Subgroup ◽  
Computer Algebra ◽  
Character Table ◽  
Computer Algebra Systems ◽  
Split Extension ◽  
Sporadic Group ◽  
Local Subgroup ◽  
Fischer Group

The Fischer group [Formula: see text] is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA.

scholarly journals Character Table of a Borel Subgroup of the Ree Groups 2F4(q2)

2009 ◽  
Vol 12 ◽  
pp. 1-53 ◽  
Author(s):  
Frank Himstedt ◽  
Shih-Chang Huang
Keyword(s):  
Borel Subgroup ◽  
Conjugacy Classes ◽  
Character Table ◽  
Irreducible Characters ◽  
Characteristic 2

AbstractWe compute the conjugacy classes and character table of a Borel subgroup of the Ree groups 2F4(22n+1) for all n ≥ 1 and prove that these Borel subgroups are M-groups. We determine the degrees of the irreducible characters of the Sylow-2-subgroups of 2F4(22n+1) and show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for 2F4(22n+1) in characteristic 2. For most of the calculations we use CHEVIE.

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 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 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.

Sign in / Sign up

Export Citation Format

Share Document