Tits chamber systems and finite group theory

The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement:Fact 1.1 (Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G.The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. The original statement stipulated one of N or G/N to be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces either N or G/N to be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem).The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a group G of finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a group H is called a π-group, where π is a set of prime numbers, if elements of H have finite orders whose prime divisors are from π. Maximal π-subgroups of a group G are called π-Hall subgroups. They exist by Zorn's lemma. Since a normal π-subgroup of G is in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.)The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider the p-Sylow for any prime p, or even the torsion part of ℂ*. Let H be this subgroup. H has a complement, but this complement is found by Zorn's Lemma (consider a maximal subgroup that intersects H trivially) and the use of Zorn's Lemma is essential. In fact, by Zorn's Lemma, any subgroup that has a trivial intersection with H can be extended to a complement of H. Since ℂ* is abelian, these complements cannot be conjugated to each other.

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BELL INEQUALITIES VIEWED IN TERMS OF EUCLIDEAN GEOMETRY AND GROUP THEORY

International Journal of Modern Physics B ◽

10.1142/s021797929500032x ◽

1995 ◽

Vol 09 (07) ◽

pp. 819-847

Author(s):

ROBERT R. TUCCI

Keyword(s):

Finite Group ◽

Group Theory ◽

Euclidean Geometry ◽

Hidden Variables ◽

Bell Inequalities ◽

Geometrical Interpretation ◽

Finite Group Theory

This paper presents a new, computer-implementable algorithm for determining Bell inequalities. The algorithm is very general: one can use it to derive the Bell inequalities for any of the experiments that are usually considered for this purpose, such as the Bohm-Bell experiment, the Clauser-Horne experiment, or the experiments with spin 1, 3/2, 2 and 5/2 particles proposed by Mermin, Schwarz and Garg. As an example, this paper applies the algorithm to the Clauser-Horne experiment. The algorithm, which is based on simple notions from n-dimensional Euclidean geometry, gives a simple geometrical interpretation to the Bell inequalities. The algorithm allows one to demonstrate that a set of Bell inequalities is complete. The algorithm uses special hidden variables that were first used by Wigner and Belinfante. This paper also shows how to use finite group theory to express a set of Bell inequalities in a form that makes its invariance under relabellings explicit.

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ARITHMETICAL SIMILARITIES: PRIME DECOMPOSITION AND FINITE GROUP THEORY (Oxford Mathematical Monographs)

Bulletin of the London Mathematical Society ◽

10.1112/s0024609399236156 ◽

1999 ◽

Vol 31 (6) ◽

pp. 744-745

Author(s):

Cheryl E. Praeger

Keyword(s):

Finite Group ◽

Group Theory ◽

Prime Decomposition ◽

Finite Group Theory

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Local Finite Group Theory

Grundlehren der mathematischen Wissenschaften - Finite Groups III ◽

10.1007/978-3-642-67997-1_1 ◽

1982 ◽

pp. 1-159 ◽

Cited By ~ 2

Author(s):

Bertram Huppert ◽

Norman Blackburn

Keyword(s):

Finite Group ◽

Group Theory ◽

Finite Group Theory

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Specific Properties of a King of Trivariable Wavelet Packets with Respect to an Integer-Valued Dilation Matrix

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.439-440.1141 ◽

2010 ◽

Vol 439-440 ◽

pp. 1141-1146

Author(s):

Jin Cang Han ◽

Yang Li

Keyword(s):

Finite Group ◽

Group Theory ◽

Frequency Analysis ◽

Matrix Theory ◽

Wavelet Packets ◽

Analysis Method ◽

Time Frequency ◽

Finite Group Theory ◽

Dilation Matrix ◽

Vector Valued

In the work, the concept of orthogonal vector-valued trivariate wavelet packets, which is a generalization of uniwavelet packets, is introduced. A new method for constructing them is developed, and their characteristics is discussed by using time-frequency analysis method, matrix theory and finite group theory. Orthogonality formulas are established.

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