Knowledge (XXG)

Diagonalizable group

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237: 278: 113: 271: 236: 302: 264: 297: 109: 136: 101: 21: 143: 50: 208: 80: 46: 248: 94: 196: 180: 76: 173: 291: 184: 244: 93:. Any closed subgroup and image of diagonalizable groups are diagonalizable. The 165: 17: 169: 29: 33: 172:
of the group. The fact plays a crucial role in the structure theory of
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of a diagonalizable group coincides with the identity component of the
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The so-called "rigidity" states that the identity component of the
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for an algebraic group. Every diagonalizable group splits over the
252: 183:(which is not necessarily compact, in contrast to a 195:. The centralizer of a maximal torus is called a 179:A connected diagonalizable group is called an 272: 8: 279: 265: 75:. This coincides with the usual notion of 49:. A diagonalizable group defined over a 104:of diagonalizable groups defined over 7: 233: 231: 97:of a diagonalizable group is dense. 71:if the isomorphism is defined over 251:. You can help Knowledge (XXG) by 14: 114:finitely generated abelian groups 235: 127:)-equivariant morphisms without 191:-torus is a torus defined over 146:and motivated the terminology. 1: 319: 230: 157:if it has no nontrivial 222:Linear algebraic groups 142:. This is an analog of 247:-related article is a 22:affine algebraic group 153:-group is said to be 161:-valued character. 112:to the category of 303:Group theory stubs 260: 259: 209:Diagonal subgroup 149:A diagonalizable 81:separable closure 47:diagonal matrices 310: 298:Algebraic groups 281: 274: 267: 239: 232: 144:Poincaré duality 95:torsion subgroup 318: 317: 313: 312: 311: 309: 308: 307: 288: 287: 286: 285: 228: 217: 205: 197:Cartan subgroup 181:algebraic torus 174:solvable groups 122: 88: 45:, the group of 44: 12: 11: 5: 316: 314: 306: 305: 300: 290: 289: 284: 283: 276: 269: 261: 258: 257: 240: 226: 225: 216: 213: 212: 211: 204: 201: 137:characteristic 120: 86: 40: 26:diagonalizable 24:is said to be 13: 10: 9: 6: 4: 3: 2: 315: 304: 301: 299: 296: 295: 293: 282: 277: 275: 270: 268: 263: 262: 256: 254: 250: 246: 241: 238: 234: 229: 223: 219: 218: 214: 210: 207: 206: 202: 200: 198: 194: 190: 186: 185:complex torus 182: 177: 175: 171: 167: 162: 160: 156: 152: 147: 145: 141: 138: 134: 131:-torsion, if 130: 126: 119: 115: 111: 107: 103: 98: 96: 92: 85: 82: 78: 74: 70: 66: 62: 59: 55: 52: 48: 43: 39: 35: 31: 27: 23: 19: 253:expanding it 245:group theory 242: 227: 221: 192: 188: 178: 163: 158: 154: 150: 148: 139: 132: 128: 124: 117: 105: 99: 90: 83: 72: 68: 64: 60: 57: 53: 41: 37: 25: 15: 166:centralizer 155:anisotropic 56:is said to 18:mathematics 292:Categories 220:Borel, A. 215:References 170:normalizer 110:equivalent 58:split over 30:isomorphic 224:, 2nd ed. 116:with Gal( 28:if it is 203:See also 102:category 34:subgroup 135:is of 243:This 187:). A 77:split 69:split 51:field 32:to a 20:, an 249:stub 100:The 108:is 89:of 63:or 36:of 16:In 294:: 199:. 176:. 280:e 273:t 266:v 255:. 193:k 189:k 159:k 151:k 140:p 133:k 129:p 125:k 123:/ 121:s 118:k 106:k 91:k 87:s 84:k 73:k 67:- 65:k 61:k 54:k 42:n 38:D

Index

mathematics
affine algebraic group
isomorphic
subgroup
diagonal matrices
field
split
separable closure
torsion subgroup
category
equivalent
finitely generated abelian groups
characteristic
Poincaré duality
centralizer
normalizer
solvable groups
algebraic torus
complex torus
Cartan subgroup
Diagonal subgroup
Stub icon
group theory
stub
expanding it
v
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e
Categories
Algebraic groups

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