Cohomology of Finite GroupsSome Historical Background This book deals with the cohomology of groups, particularly finite ones. Historically, the subject has been one of significant interaction between algebra and topology and has directly led to the creation of such important areas of mathematics as homo logical algebra and algebraic K-theory. It arose primarily in the 1920's and 1930's independently in number theory and topology. In topology the main focus was on the work ofH. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among others) made significant contributions. The main thrust of the early work here was to try to understand the meanings of the low dimensional homology groups of a space X. For example, if the universal cover of X was three connected, it was known that H2(X; A. ) depends only on the fundamental group of X. Group cohomology initially appeared to explain this dependence. In number theory, group cohomology arose as a natural device for describing the main theorems of class field theory and, in particular, for describing and analyzing the Brauer group of a field. It also arose naturally in the study of group extensions, N |
Contents
I | 1 |
II | 7 |
III | 8 |
IV | 12 |
V | 14 |
VI | 16 |
VII | 17 |
VIII | 19 |
LXV | 173 |
LXVI | 174 |
LXVII | 175 |
LXVIII | 177 |
LXIX | 178 |
LXX | 179 |
LXXI | 180 |
LXXII | 183 |
IX | 22 |
X | 24 |
XI | 26 |
XII | 30 |
XIII | 32 |
XIV | 34 |
XVI | 35 |
XVII | 36 |
XVIII | 38 |
XIX | 40 |
XX | 43 |
XXI | 50 |
XXII | 53 |
XXIV | 54 |
XXV | 55 |
XXVI | 64 |
XXVII | 66 |
XXVIII | 68 |
XXIX | 70 |
XXX | 73 |
XXXI | 78 |
XXXII | 83 |
XXXIII | 89 |
XXXIV | 95 |
XXXV | 102 |
XXXVI | 104 |
XXXVII | 108 |
XXXVIII | 111 |
XL | 112 |
XLI | 115 |
XLII | 116 |
XLIII | 117 |
XLIV | 119 |
XLV | 122 |
XLVII | 125 |
XLVIII | 128 |
XLIX | 131 |
L | 137 |
LI | 139 |
LII | 140 |
LIV | 141 |
LV | 142 |
LVI | 146 |
LVII | 150 |
LVIII | 152 |
LIX | 153 |
LX | 155 |
LXI | 157 |
LXII | 161 |
LXIII | 166 |
LXIV | 171 |
LXXIII | 190 |
LXXIV | 194 |
LXXV | 196 |
LXXVI | 199 |
LXXVII | 203 |
LXXVIII | 208 |
LXXIX | 213 |
LXXX | 214 |
LXXXI | 221 |
LXXXII | 225 |
LXXXIII | 228 |
LXXXIV | 234 |
LXXXV | 238 |
LXXXVI | 245 |
LXXXVII | 246 |
LXXXVIII | 247 |
LXXXIX | 248 |
XCI | 256 |
XCII | 260 |
XCIV | 262 |
XCV | 265 |
XCVI | 267 |
XCVIII | 270 |
XCIX | 273 |
C | 275 |
CI | 277 |
CII | 278 |
CIII | 279 |
CIV | 280 |
CV | 281 |
CVI | 282 |
CVII | 283 |
CVIII | 287 |
CIX | 288 |
CX | 290 |
CXI | 292 |
CXII | 294 |
CXIII | 295 |
CXIV | 296 |
CXV | 297 |
CXVI | 298 |
CXVII | 301 |
CXIX | 304 |
CXX | 306 |
CXXI | 307 |
CXXII | 311 |
CXXIII | 312 |
| 315 | |
| 321 | |
Other editions - View all
Common terms and phrases
2-subgroup abelian action acyclic Ap(G automorphism Brauer group calculation central simple Chap classifying space coefficients cohomology groups commutative complex conjugacy classes conjugate Consequently construction Corollary coset CW complex cyclic d₁ defined denote determined diagram dimension dimensional division algebra e₁ elementary elements exact sequence example extension F-algebra fiber fibration finite group follows G₁ Gal(K/F Galois given group cohomology group G Hence homology homomorphism homotopy Hoont Hopf algebra inclusion induced injective isomorphism kernel Lemma Let G maximal module Moreover multiplication non-trivial non-zero normal obtain odd prime p-group Poincaré series Proof quaternion quaternion group Quillen quotient Remark resolution restriction map result simple groups spectral sequence Sq¹ Steenrod Steenrod algebra structure subgroup summand suppose surjective symmetric groups tensor Theorem topology trivial wreath product zero σ₁ Στ
Popular passages
Page 320 - T. Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Mathematics 397 (Springer, Berlin, Heidelberg, New York, 1974).
Page 319 - Milgram. On the moduli space of SU(n) monopoles and holomorphic maps to flag manifolds, J. Diff. Geom. 38 (1993), 39-103.
Page 319 - J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67 (1958), 150-171. [Mi2] J. Milnor, Groups which act on 5
Page 320 - JP. Serre. Cohomologie modulo 2 des complexes d' Eilenberg-MacLane, Comm. Math. Helv., 27 (1953), 198 232.
Page 319 - T. NAKAYAMA, On modules of trivial cohomology over a finite group, I, Illinois J. Math.
Page 320 - Math. 69 (1959), 700-712. [RSY] A. Ryba, S. Smith, S. Yoshiara, Some projective modules determined by sporadic geometries, UIC preprint (1988).