scholarly journals Mac Lane Method for Determination of Extensions of Finite Groups. II. An Example for the Dihedral Group D2

1992 ◽  
Vol 82 (3) ◽  
pp. 395-406 ◽  
Author(s):  
T. Lulek ◽  
R. Chatterjee
Keyword(s):  
Finite Groups ◽  
Dihedral Group ◽  
Mac Lane

scholarly journals The Laplacian Energy of Conjugacy Class Graph of Some Finite Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (1) ◽  
pp. 59-65
Author(s):  
Rabiha Mahmoud ◽  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Laplacian Eigenvalues ◽  
Laplacian Matrices ◽  
Laplacian Energy ◽  
The Difference ◽  
Generalized Quaternion ◽  
Class Graph

Let G be a dihedral group and its conjugacy class graph. The Laplacian energy of the graph, is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined.

Notiz: Quantification of Symmetry

1996 ◽  
Vol 51 (7) ◽  
pp. 882-883
Author(s):  
Igor Novak
Keyword(s):  
Finite Groups ◽  
Group Element ◽  
Symmetry Point ◽  
Point Groups ◽  
New Criterion

Abstract A new mathematical criterion is suggested for symmetry ranking, i.e. determination of an “absolute symmetry scale” for discrete, finite groups. The criterion is based on both, the periods (orders) of each group element and the order of the group itself. This is different from the current criteria which consider only the orders of the groups themselves. The symmetry ranking, based on the new criterion, is applied to the symmetry point groups.

Suatu Hubungan Kumpulan Pusat–2 dengan Teori Kebarangkalian

2012 ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Hasimah Sapiri
Keyword(s):  
Probability Theory ◽  
Finite Group ◽  
Finite Groups ◽  
Random Element ◽  
Symmetric Groups ◽  
Finite Rings ◽  
Nonabelian Group ◽  
Central Group ◽  
Upper Limit

Penentuan darjah keabelanan bagi suatu kumpulan tak abelan telah diperkenalkan untuk kumpulan simetri oleh Erdos dan Turan [1]. Dalam tahun 1973, Gustafson [2] mengkajinya bagi kumpulan terhingga sementara MacHale [3] mengkajinya bagi gelanggang terhingga dalam tahun 1976. Dalam kajian ini, beberapa keputusan yang berkaitan dengan Pn(G), kebarangkalian bahawa suatu unsur rawak dengan kuasa ke–n dalam suatu kumpulan pusat–2 G adalah kalis tukar tertib dengan unsur rawak yang lain dalam kumpulan yang sama, akan diberikan. Seterusnya, batas atas bagi P2(G) diperoleh. Kata kunci: Teori kebarangkalian, teori kumpulan, kumpulan terhingga, kalis tukar tertib The determination of the abelianness of a nonabelian group has been introduced for symmetric groups by Erdos & Turan [1]. In 1973, Gustafson [2] did this research for the finite groups while MacHale [3] determined the abelianness for finite rings in 1976. In this research, some results on Pn(G), the probability that the n–th power of a random element in a 2–central group G commutes with another random element from the same group, will be presented. Furthermore, the upper limit of P2(G) is obtained. Key words: Probability theory, group theory, finite group, commutative

scholarly journals Translation planes of odd order and odd dimension

1979 ◽  
Vol 2 (2) ◽  
pp. 187-208 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Translation Planes ◽  
Solvable Subgroup ◽  
Joint Paper ◽  
Desarguesian Planes ◽  
Odd Order ◽  
Collineation Groups ◽  
Zero Vector

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.The paper is concerned with the case where the plane is defined on a vector space of dimension2d overGF(q), whereqanddare odd. If the stabilizer of the zero vector is non-solvable, letG0be a minimal normal non-solvable subgroup. We suspect thatG0must be isomorphic to someSL(2,u)or homomorphic toA6orA7. Our main result is that this is the case whendis the product of distinct primes.The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow2-groups whendandqare both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order27(i.e.,dandqare both equal to3) which admitsSL(2,13).

scholarly journals The Grothendieck--Teichm\"{u}ller group of~PSL(2,q)

2018 ◽  
Vol 21 (2) ◽  
pp. 241-251
Author(s):  
Pierre Guillot
Keyword(s):  
Finite Groups ◽  
Dihedral Group ◽  
Monodromy Group ◽  
Derived Length

AbstractIn this paper, we show that the Grothendieck–Teichmüller group of{\operatorname{PSL}(2,q)}, or more precisely the group{\mathcal{G\kern-0.569055ptT}_{\kern-1.707165pt1}(\operatorname{PSL}(2,q))}as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, whenqis even, we show that it is trivial. We explain how it follows that the moduli field of any “dessin d’enfant” whose monodromy group is{\operatorname{PSL}(2,q)}has derived length{\leq 3}. This paper can serve as an introduction to the general results on the Grothendieck–Teichmüller group of finite groups obtained by the author.

Finite quasisimple groups of 2×2 matrices over a division ring

1985 ◽  
Vol 97 (3) ◽  
pp. 415-420
Author(s):  
B. Hartley ◽  
M. A. Shahabi Shojaei
Keyword(s):  
Finite Groups ◽  
Division Ring ◽  
Characteristic Zero ◽  
Multiplicative Group ◽  
Main Tool ◽  
Number Fields ◽  
Soluble Groups ◽  
Trivial Element ◽  
Difficult Part

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

Finite Rotation Groups in Low Dimensions

1968 ◽  
Vol 20 ◽  
pp. 711-719 ◽  
Author(s):  
Donald K. Friesen
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Rotation Group ◽  
Finite Rotation ◽  
Low Dimensions ◽  
Rotation Groups

Let F be a vector space of dimension two, three, or four over a field of characteristic not two, and let V have a non-singular orthogonal metric. The problem discussed in this paper is the determination of all finite groups that can occur as subgroups of the rotation group of V.

scholarly journals AUTOMORPHISM CRITERIA FOR M*-GROUPS

2004 ◽  
Vol 47 (2) ◽  
pp. 339-351 ◽  
Author(s):  
Emilio Bujalance ◽  
Francisco-Javier Cirre ◽  
Peter Turbek
Keyword(s):  
Finite Groups ◽  
Commutator Subgroup ◽  
Subject Classification ◽  
Direct Products ◽  
Primary 20D45

AbstractWe prove that the determination of all $M^*$-groups is essentially equivalent to the determination of finite groups generated by an element of order 3 and an element of order 2 or 3 that admit a particular automorphism. We also show how the second commutator subgroup of an $M^*$-group $G$ can often be used to construct $M^*$-groups which are direct products with $G$ as one factor. Several applications of both methods are given.AMS 2000 Mathematics subject classification: Primary 20D45; 20E36. Secondary 14H37; 30F50

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