The Relationship Between Finite Groups and Completely Orthogonal Squares, Cubes, and Hyper-Cubes

AbstractFor a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

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Character theory of infinite wreath products

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.1365 ◽

2005 ◽

Vol 2005 (9) ◽

pp. 1365-1379 ◽

Cited By ~ 3

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.

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Conformality and p-isomorphism in finite nilpotent groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700005541 ◽

1967 ◽

Vol 7 (2) ◽

pp. 165-171 ◽

Cited By ~ 4

Author(s):

C. D. H. Cooper

Keyword(s):

Finite Groups ◽

Abelian Groups ◽

Nilpotent Groups ◽

Equivalence Relations ◽

The Relationship

This paper discusses the relationship between two equivalence relations on the class of finite nilpotent groups. Two finite groups are conformal if they have the same number of elements of all orders. (Notation: G ≈ H.) This relation is discussed in [4] pp 107–109 where it is shown that conformality does not necessarily imply isomorphism, even if one of the groups is abelian. However, if both groups are abelian the position is much simpler. Finite conformal abelian groups are isomorphic.

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A Note on the p-Deficiency Class of a Finite Group

Algebra Colloquium ◽

10.1142/s1005386712000235 ◽

2012 ◽

Vol 19 (02) ◽

pp. 353-358 ◽

Cited By ~ 2

Author(s):

Tianze Li ◽

Weigang Xu ◽

Jiping Zhang

Keyword(s):

Finite Group ◽

Solvable Group ◽

Finite Groups ◽

Characteristic P ◽

P Deficiency ◽

Class 1 ◽

A Finite Group ◽

P Type ◽

The Relationship

In this note, we explore the relationship between finite groups of characteristic p type and those of p-deficiency class 1. We study the structure of finite groups of characteristic p type. Besides, we show that the p-rank (resp., p-length) of a p-solvable group which is of exact p-deficiency class r(> 0) is bounded by r (resp., a function of r).

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THE RELATIONSHIP BETWEEN FINITE GROUPS AND COMPLETELY ORTHOGONAL SQUARES, CUBES, AND HYPER-CUBES

Biometrika ◽

10.1093/biomet/35.3-4.277 ◽

1948 ◽

Vol 35 (3-4) ◽

pp. 277-282 ◽

Cited By ~ 3

Author(s):

K. A. BROWNLEE ◽

P. K. LORAINE

Keyword(s):

Finite Groups ◽

The Relationship

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Conway and iteration hemirings Part 1

International Journal of Algebra and Computation ◽

10.1142/s0218196714500210 ◽

2014 ◽

Vol 24 (04) ◽

pp. 461-482

Author(s):

M. Droste ◽

Z. Ésik ◽

W. Kuich

Keyword(s):

Finite Groups ◽

Transition Systems ◽

Free Constructions ◽

The Relationship

Conway hemirings are Conway semirings without a multiplicative unit. We also define iteration hemirings as Conway hemirings satisfying certain identities associated with the finite groups. Iteration hemirings are iteration semirings without a multiplicative unit. We provide an analysis of the relationship between Conway hemirings and (partial) Conway semirings and describe several free constructions. In the second part of the paper we define and study hemimodules of Conway and iteration hemirings, and show their applicability in the analysis of quantitative aspects of the infinitary behavior of weighted transition systems. These include discounted and average computations of weights investigated recently.

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Local Freeness of Profinite Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1982-063-7 ◽

1982 ◽

Vol 25 (4) ◽

pp. 441-446

Author(s):

Andrew Pletch

Keyword(s):

Finite Groups ◽

Solvable Groups ◽

Nilpotent Groups ◽

Profinite Groups ◽

Local Properties ◽

The Relationship

AbstractIn this paper we discuss the relationship between local properties such as freeness and projectivity of a group and the freeness or projectivity of its pro-C-completion. We show that for certain classes, C, of finite groups (e.g. p-groups, nilpotent groups, super-solvable groups) the pro-C-completion of a locally free pro-C-group is a free pro-C-group. We also show that under certain circumstances the converse is also true but we leave open the question, for example, of whether a locally free pro-p-group is free.

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The relationship between finite groups and Clifford algebras

Journal of Mathematical Physics ◽

10.1063/1.526260 ◽

1984 ◽

Vol 25 (4) ◽

pp. 738-742 ◽

Cited By ~ 22

Author(s):

Nikos Salingaros

Keyword(s):

Finite Groups ◽

Clifford Algebras ◽

The Relationship

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Burnside-Brauer Theorem for Table Algebras

The Electronic Journal of Combinatorics ◽

10.37236/691 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 1

Author(s):

J. Bagherian ◽

A. Rahnamai Barghi

Keyword(s):

Finite Groups ◽

Character Theory ◽

Table Algebras ◽

Table Algebra ◽

The Relationship

In the character theory of finite groups the Burnside-Brauer Theorem is a well-known result which deals with products of characters in finite groups. In this paper, we first define the character products for table algebras and then by observing the relationship between the characters of a table algebra and the characters of its quotient, we provide a condition in which the products of characters of table algebras are characters. As a main result we state and prove the Burnside-Brauer Theorem on finite groups for table algebras.

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