Clebsch–Gordan decomposition for transitive representations

1985 ◽  
Vol 63 (8) ◽  
pp. 1061-1064 ◽  
Author(s):  
B. Lulek ◽  
T. Lulek ◽  
R. Chatterjee ◽  
J. Biel
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Fibre Bundle ◽  
Canonical Realization ◽  
A Finite Group ◽  
Method Of Evaluation ◽  
Transitive Representation

The method of evaluation of Clebsch–Gordan coefficients to calculate the direct product of transitive representations for a finite group is developed. This procedure is based on the Mackey theorem and on the canonical realization of transitive representations. It is shown that the Clebsch–Gordan decomposition is associated with a natural interpretation of an orbit of a resultant transitive representation and it is analogous to a fibre bundle with the fibration being determined by constituent transitive representations.

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

Some Remarks on Groups Admitting a Fixed-Point-Free Automorphism

1968 ◽  
Vol 20 ◽  
pp. 1300-1307 ◽  
Author(s):  
Fletcher Gross
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Simple Group ◽  
Identity Element ◽  
Simple Groups ◽  
Cyclic Groups ◽  
A Finite Group ◽  
Open Question

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.

A Remark on the Units of Finite Order in The Group Ring of a Finite Group

1974 ◽  
Vol 17 (1) ◽  
pp. 129-130 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Finite Group ◽  
Finite Order ◽  
Direct Product ◽  
Group Ring ◽  
Integral Group Ring ◽  
Integral Group ◽  
Quaternion Group ◽  
Group Of Units ◽  
A Finite Group

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will proveLet G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

The group J4 × J4 is recognizable by spectrum

2020 ◽  
pp. 2150061
Author(s):  
Ilya B. Gorshkov ◽  
Natalia V. Maslova
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Element Orders ◽  
Sporadic Group ◽  
A Finite Group

The spectrum of a finite group is the set of its element orders. In this paper, we prove that the direct product of two copies of the finite simple sporadic group [Formula: see text] is uniquely determined by its spectrum in the class of all finite groups.

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.

Integral Representations of the Direct Product of Groups

1963 ◽  
Vol 15 ◽  
pp. 625-630 ◽  
Author(s):  
Alfredo Jones
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Principal Ideal ◽  
Integral Representations ◽  
Principal Ideal Domain ◽  
Linear Combinations ◽  
Quotient Field ◽  
A Finite Group ◽  
Torsion Free ◽  
Product Of Groups

Let G be a finite group and R a Dedekind domain with quotient field K. We denote by RG the group ring of formal linear combinations of elements of G with coefficients in R. By an RG-module we understand a unital left RG-module which is finitely generated and torsion-free as R-module. In particular, if R is a principal ideal domain this is equivalent to considering representations of G by matrices with entries in R.

Blocks of defect zero in finite groups with conjugate subgroups of a given prime order

2017 ◽  
Vol 16 (11) ◽  
pp. 1750217
Author(s):  
Tianze Li ◽  
Yanjun Liu ◽  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Groups ◽  
Prime Order ◽  
Text Block ◽  
Order Group ◽  
Odd Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] be a prime. In this note, we show that if [Formula: see text] and all subgroups of [Formula: see text] of order [Formula: see text] are conjugate, then either [Formula: see text] has a [Formula: see text]-block of defect zero, or [Formula: see text] and [Formula: see text] is a direct product of a simple group [Formula: see text] and an odd order group. This improves one of our previous works.

A Lower Bound for the Real Genus of a Finite Group

1994 ◽  
Vol 46 (06) ◽  
pp. 1275-1286 ◽  
Author(s):  
Coy L. May
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Direct Product ◽  
Dihedral Group ◽  
Surface Group ◽  
Crystallographic Group ◽  
Klein Surface ◽  
Standard Representation ◽  
The Real ◽  
A Finite Group

Abstract Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we obtain a good general lower bound for the real genus of the group G. We use the standard representation of G as a quotient of a non-euclidean crystallographic group by a bordered surface group. The lower bound is used to determine the real genus of several infinite families of groups; the lower bound is attained for some of these families. Among the groups considered are the dicyclic groups and some abelian groups. We also obtain a formula for the real genus of the direct product of an elementary abelian 2-group and an “even” dicyclic group. In addition, we calculate the real genus of an abstract family of groups that includes some interesting 3-groups. Finally, we determine the real genus of the direct product of an elementary abelian 2-group and a dihedral group.

scholarly journals Algebraic Properties of Implication-Based Anti-Fuzzy Subgroup over a Finite Group

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1116-1120
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Algebraic Properties ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
A Finite Group ◽  
Fuzzy Direct Product

This article deals with few algebraic characteristics of implication-based anti-fuzzy subgroup of a finite group.In addition, the implication-based anti-fuzzy direct product of implication-based anti-fuzzy subgroups over finite groups is developed and studied elaborately. The condition for an implication-based anti-fuzzy subgroup of a finite group to be a conjugate to another implication-based anti-fuzzy subgroup is conceptualized. Some of their characteristics are investigated in this paper.

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