群论导论-第4版 内容简介

Group Theory is a vast subject and, in this Introduction （as well as in theearlier editions）, I have tried to select important and representative theoremsand to organize them in a coherent way. Proofs must be clear, and examplesshould illustrate theorems and also explain the presence of restrictive hypo-theses. ！ also believe that some history should be given so that one canunderstand the origin of problems and the context in which the subjectdeveloped. Just as each of the earlier editions differs from the previous one in a signifi-cant way, the present （fourth） edition is genuinely different from the third.Indeed, this is already apparent in the Table of Contents. The book nowbegins with the unique factorization of permutations into disjoint cycles andthe parity of permutations; only then is the idea of group introduced. This isconsistent with the history of Group Theory, for these first results on permu-tations can be found in an 1815 paper by Cauchy, whereas groups of permu-tations were not introduced until 1831 （by Galois）But even if history

群论导论-第4版 本书特色

Group Theory is a vast subject and, in this Introduction (as well as in theearlier editions), I have tried to select important and representative theoremsand to organize them in a coherent way. Proofs must be clear, and examplesshould illustrate theorems and also explain the presence of restrictive hypo-theses. I also believe that some history should be given so that one canunderstand the origin of problems and the context in which the subject developed.

群论导论-第4版 目录

preface to the fourth edition
from preface to the third edition
to the reader
chapter 1groups and homomorphisms
permutations
cycles
factorization into disjoint cycles
even and odd permutations
semigroups
groups
homomorphisms
chapter 2the isomorphism theorems
subgroups
lagrange's theorem
cyclic groups
normal subgroups
quotient groups
the isomorphism theorems
correspondence theorem
direct products
chapter 3symmetric groups and g-sets
conjugates
symmetric groups
the simplicity of a.
some representation theorems
g-sets
counting orbits
some geometry
chapter 4the sylow theorems
p-groups
the sylow theorems
groups of small order
chapter 5normal series
some galois theory
the jordan-ho1der theorem
solvable groups
two theorems of p. hall
central series and nilpotent groups
p-groups
chapter 6finite direct products
the basis theorem
the fundamental theorem of finite abelian groups
canonical forms; existence
canonical forms; uniqueness
the kruli-schmidt theorem
operator groups
chapter 7extensions and cohomology
the extension problem
automorphism groups
semidirect products
wreath products
factor sets
theorems of schur-zassenhaus and gaschijtz
transfer and burnside's theorem
projective representations and the schur multiplier
derivations
chapter 8some simple linear groups
……
chapter 9permutations and the mathieu groups
chapter 10abelian groups
chapter 11free groups and free products
chapter 12the word problem
epilogue
bibliography
notation
index