Representations and Characters of Groups, Second Edition - Gordon James and Martin Liebeck Index More Information

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Cambridge University Press 0521812054 - Representations and Characters of Groups, Second Edition - Gordon James and Martin Liebeck Index More information

Index

A4, 112, 130, 136, 181, 308 permutation, 129 A5, 10, 112, 116, 220, 312, 359 product, 176, 192 A6, 116, 222, 354, 360 real, 263 A7, 223 realized over R, 265 An, 5, 9, 11, 111, 343 reducible, 119 abelian group, 3, 11, 81, 82 regular, 127, 128, 150 action, 337 trivial, 122 algebra, 55, 56 character table, 159 algebraic integer, 244, 362 A4, 181 algebraic number, 361 A5, 221, 359 alternating group, 5, 9, 11, 111 A6, 359, 424 antisymmetric part, 196, 273 C2, 160 associative, 2 C3, 160 C4, 412 basis, 15 C2 3 C2, 415 natural, 45, 54 Cn,82 bijection, 6 D6, 160 bilinear form, 269 D8, 161 symmetric, 269 D10, 415 skew-symmetric, 269 D12 S3 3 C2, 207, 419 Brauer±Fowler Theroem, 278 D2n (n odd), 182 Burnside's Lemma, 340 D2n (n even), 183 Burnside's Theorem, 363, 364 D6 3 D6, 423 E, of order 18, 434 C,2 F7,3, 240, 417 Cn, 2, 82, 88 F11,5,433 centralizer, 106 Fp,q, 291 centre GL(2,q), 327 of group, 85, 107, 116, 298 PSL(2,7), 318 of group algebra, 83, 114, 153 PSL(2,8), 445 change of basis, 24 PSL(2,11), 443 character, 118 Q8, 416 degree, 122, 247 S4, 180 faithful, 125, 195 S5, 201, 262 generalized, 355 S6, 205 induced, 230, 234, 236 SL(2,3), 440 integer-valued, 253 SL(2,7), 442 irreducible, 119 SL(2,q), 445 kernel of, 125 T12, 186 linear, 122, 172, 174 T4n, 420

455

456 Representations and characters of groups

U6n, 421 FG-submodule, 49 V24, 422 FG-homomorphism, 61 V8n, 421 FG-isomorphism, 63 direct product, 206 Frobenius group, 290 order 16, 305, 306, 307 Frobenius Reciprocity Theorem, 232 order 27, 435 Frobenius±Schur Count of Involutions, 277 order , 32, 308 function, 6 order p3,301 bijective, 6 order pq, 291 injective, 6 p-group, 300 invertible, 6 class algebra constants, 349 surjective, 6 class equation, 107 class function, 152 GL(n,F), 3 class sum, 114 GL(2,q), 324, 343 Clifford's Theorem, 216 general linear group, 3 complete set, 101 group, 1 completely reducible, 74 abelian, 3, 11, 81, 82 composition, 2 alternating, 5, 9, 11, 111 composition factor, 90 cyclic, 2, 4, 12, 82, 88 common, 91, 96 dicyclic, 178 congruences, 259 dihedral, 2, 12, 107, 181 conjugacy class, 104 factor, 9 conjugate, 104, 361 ®nite, 2 constituent, 143, 213 general linear, 3 coset, 8 order, 2 cycle notation, 8 order p3, 302, 304 cycle-shape, 109 orthogonal, 367 cyclic group, 2, 4, 12, 82, 88 projective special linear, 312 quaternion, 5 D2n, 2, 12, 107, 181 rotation, 368 degree, 30, 122, 249 simple, 10, 250, 278, 311, 318, 353, derived subgroup, 173 363, 364 diagonalization, 83 soluble, 365 dicyclic group T4n, 178, 187, 281, 420 special linear, 311 dihedral group, 2, 12, 107, 181 symmetric, 3, 109, 116, 175, 254 dimension, 15 symmetry, 368 direct product, 5, 206 group algebra, 55 direct sum, 17, 66 external, 18 H < G,3 H v G,9 eigenvalue, 24 HomCG(V, W ), 95, 96 eigenvector, 24 homogeneous component, 376 endomorphism, 20 homomorphism, 6, 10, 61 equivalent, 32, 46 even permutation, 5 ideal, 256 expansion±contraction mode, 381 maximal, 257 external direct sum, 18 proper, 257 index of subgroup, 9 F R or C,3 indicator function, 273 Fn,15 induced character, 230, 234, 236 Fp,q, 290 induced module, 226, 228 FG,53 inner product, 134 factor group, 9 involution, 277, 353 faithful character, 125, 195 irreducible character, 119 faithful module, 44, 56, 85 irreducible module, 50, 74, 79, 91 faithful representation, 34 irreducible representation, 50, 79 FG-module, see module isomorphism, 7, 20, 63

Index 457

kernel, 10, 19, 34, 124, 125 projection, 27, 67 projective special linear group, 312 Lagrange's Theorem, 9 lift, 169 Q8, 5, 116, 177, 278, 416 linear character, 122, 173, 174 quaternion group, 5, 116, 177, 278, 416 linear transformation, 18 linearly dependent, 15 R,3 linearly independent, 15 rank, 342 Rank±Nullity Theorem, 19 real character, 263 Maschke's Theorem, 70, 76 real conjugacy class, 263 matrix, 21 real element, 263 change of basis, 24 reducible character, 119 diagonal, 26 reducible module, 50 identity, 3, 21 reducible representation, 50 invertible, 23 regular character, 127, 128, 150 permutation, 45 regular module, 56 methane, 384 regular representation, 56 minimal polynomial, 361 representation, 30 module, 39 degree, 30, 249 completely reducible, 74 equivalent, 32, 46 faithful, 44, 56, 85 faithful, 34 irreducible, 50, 79, 85 irreducible, 50, 79 permutation, 45, 62 kernel of, 34, 124 reducible, 50 reducible, 50 regular, 56 regular, 56 trivial, 43 trivial, 34 representatives, 105 natural basis, 45, 54 restriction, 210 normal modes of vibration, 372, 373 rotation group, 368 normal p-complement, 251 rotation mode, 379 normal subgroup, 9, 113, 171, 215, 216, rotation submodule, 380, 394 217 S4, 44, 110, 113, 180, 275 S , 111, 201, 262 odd permutation, 5 5 S , 116, 205 orbit, 338 6 S , 223 order of G,2 7 S , 3, 109, 116, 175, 254, 343, 344 order of g,4 n SL(2,3), 319, 440 orthogonal group, 367 SL(2,7), 320, 442 orthogonality relations, 161 SL(2, p), 311 SL(2,q), 336, 445 PSL(2,7), 312, 318, 319, 354, 359, 360 Schur's Lemma, 78 PSL(2,11), 321, 359 simple group, 10, 250, 278, 311, 318, PSL(2, p), 312 363, 364 p-group, 298 skew-symmetric bilinear form, 269 p9-part, 256, 258 special linear group, 311 permutation, 3, 5 stabilizer, 339 even, 5 subgroup, 3, 4 odd, 5 cyclic, 4 permutation module, 45, 62, 340 derived, 173 permutation character, 129, 340 generated, 4 permutation matrix, 45 normal, 9, 113, 171, 215, 216, 217 powers of characters, 193 submodule, 49 presentation, 3 irreducible, 74 primitive root, 284 Sylow's Theorem, 354, 365 product of characters, 176, 192 symmetric bilinear form, 269

458 Representations and characters of groups

symmetric group, 3, 109, 116, 175, 254 trivial character, 122 symmetric part, 196, 273 module, 43 symmetry group, 368 representation, 34

T4n, 178, 187, 281, 420 U6n, 178, 187, 421 tensor product module, 190 tensor product space, 188 V8n, 178, 187, 421 trace, 117 Vandermonde matrix, 194 transitive, 338, 341 vibratory modes, 381 transitivity of induction, 229 translation mode, 379 water, 369, 374 translation submodule, 380, 394 transposition, 5 Z,2