Theory of Groups of Finite Order

by William Burnside

Overview

A classic introduction to group theory for nearly a century, this is the book most often cited in texts on group theory for detailed expositions of basic concepts.

After introducing permutation notation and defining group, the author discusses the simpler properties of group that are independent of their modes of representation; composition-series of groups; isomorphism of a group with itself; Abelian groups; groups whose orders are the powers of primes; and Sylow's theorem. Permutation groups and groups of linear substitutions receive an extensive treatment; two chapters are devoted to the graphical representation of groups, and the closing chapter to congruence groups. Forty-five pages of notes at the back of the book offer ample treatment of special topics.

Product Details

ISBN-13:	9789357949699
Publisher:	Alpha Edition
Publication date:	12/18/2023
Pages:	446
Product dimensions:	8.50(w) x 11.00(h) x 0.90(d)

Table of Contents

Chapter I.	On Permutations
Chapter II.	The Definition of a Group
Chapter III.	On the Simpler Properties of a Group Which Are Independent of Its Mode of Representation
Chapter IV.	Further Properties of a Group Which Are Independent of Its Mode of Representation
Chapter V.	On the Composition-Series of a Group
Chapter VI.	On the Isomorphism of a Group with Itself
Chapter VII.	On Abelian Groups
Chapter VIII.	On Groups Whose Orders Are the Powers of Primes
Chapter IX.	On Sylow's Theorem
Chapter X.	On Permutation-Groups: Transitive and Intransitive Groups: Primitive and Imprimitive Groups
Chapter XI.	On Permutation-Groups: Transitivity and Primitivity: (Concluding Properties)
Chapter XII.	On the Representation of a Group of Finite Order as a Permutation-Group
Chapter XIII.	On Groups of Linear Substitutions; Reducible and Irreducible Groups
Chapter XIV.	On the Representation of a Group of Finite Order as a Group of Linear Substitutions
Chapter XV.	On Group-Characteristics
Chapter XVI.	Some Applications of the Theory of Groups of Linear Substitutions and of Group-Characteristics
Chapter XVII.	On the Invariants of Groups of Linear Substitutions
Chapter XVIII.	On the Graphical Representation of a Group
Chapter XIX.	On the Graphical Representation of Groups: Groups of Genus Zero and Unity: Cayley's Colour-Groups
Chapter XX.	On Congruence Groups
Note A.	On the equation N = h[subscript 1] + h[subscript 2] + ... + h[subscript r]	461
Note B.	On the group of isomorphisms of a group	463
Note C.	On the symmetric group	464
Note D.	On the completely reduced form of a group of monomial substitutions	470
Note E.	On the irreducible representations of a group which has a self-conjugate sub-group of prime index	472
Note F.	On groups of finite order which are simply isomorphic with irreducible groups of linear substitutions	476
Note G.	On the representation of a group of finite order as a group of linear substitutions with rational coefficients	479
Note H.	On the group of the twenty-seven lines on a cubic surface	485
Note I.	On the conditions of reducibility of a group of linear substitutions of finite order	489
Note J.	On conditions for the finiteness of the order of a group of linear substitutions	491
Note K.	On the representation of a group of finite order as a group of birational transformations of an algebraic curve	496
Note L.	On the group-characteristics of the fractional linear group	499
Note M.	On groups of odd order	503
Note N.	On the orders of simple groups	504
Note O.	On algebraic numbers	505
	Index of technical terms	507
	Index of Authors Quoted	508
	General Index	509