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 | |
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Common terms and phrases
2-subgroup action acyclic automorphism Brauer group Bz/p calculation central simple Chap classifying space coefficients cohomology groups cohomology ring commutative complex conjugacy classes conjugate Consequently construction Corollary coset CW complex cyclic defined denote determined diagram dimension dimensional division algebra elementary elements exact sequence example F-algebra F₂ fiber fibration finite group follows Gal(K/F Galois given group cohomology group G Hence homology homomorphism homotopy Hont Hopf algebra inclusion induced injective isomorphism kernel Lemma Let G maximal module Moreover multiplication non-trivial non-zero Note obtain odd prime p-group Poincaré series poset Proof quaternion quaternion group Quillen quotient Remark resolution restriction map result ring of invariants simple groups SP2n spectral sequence sporadic groups Sq¹ Steenrod algebra structure summand suppose surjective symmetric groups tensor Theorem topology trivial wreath product zero
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).