Download e-book for kindle: Algebraic Generalizations of Discrete Groups: A Path to by Benjamin Fine

By Benjamin Fine

ISBN-10: 0824703197

ISBN-13: 9780824703196

A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic learn of Fuchsian teams to the extra common context of one-relator items and similar team theoretical concerns. It offers a self-contained account of definite traditional generalizations of discrete teams.

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Additional resources for Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products

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REPRESENTATIONS (1) F is of first kind, (2) Γ has integral character, (3) Γ is irreducible. A result on the imprimitivity of matrix groups is the following. A matrix group is imprimitive if it contains a normal abelian subgroup which is not contained in the center. Let us conclude our study of matrix groups with a listing of some further properties of characters. The first property is shown by an equation for the product of two values of the character of a matrix group: x(D1)x(D2) = ^¿xiD^DzDJ.

Theorem. The dimension of an irreducible matrix group of second kind must be even. It is well known that time reversal in quantum mechanics is intimately connected with complex conjugation, and it is occasionally useful in group theory to introduce a time reversal operator. If Γ is a matrix group of first or second kind, then a unitary matrix U such that Γ* = UTU'1 is called a time reversal operator associated with the matrix group Γ. Let us call a matrix group integral if it is equivalent to a matrix group in which all matrices are integral, and rational if it is equivalent to a matrix group in which every matrix element is a rational number.

The dimension of an irreducible matrix group of second kind must be even. It is well known that time reversal in quantum mechanics is intimately connected with complex conjugation, and it is occasionally useful in group theory to introduce a time reversal operator. If Γ is a matrix group of first or second kind, then a unitary matrix U such that Γ* = UTU'1 is called a time reversal operator associated with the matrix group Γ. Let us call a matrix group integral if it is equivalent to a matrix group in which all matrices are integral, and rational if it is equivalent to a matrix group in which every matrix element is a rational number.

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Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products by Benjamin Fine


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