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Direct limit of groups

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surprising, given that they are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations.
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Certain examples of stable groups are easier to study than "unstable" groups, the groups occurring in the limit. This is
83: 390: 116: 291: 54: 65: 50: 358: 259: 447: 452: 161:. They are sometimes called stable groups, though this term normally means something quite different in 90: 442: 35: 39: 154: 185: 352: 8: 158: 146: 394: 348: 150: 344: 337: 329: 181: 436: 421: 412: 333: 406: 162: 138: 130: 249:{\displaystyle \operatorname {GL} (n,A)\to \operatorname {GL} (n+1,A)} 24: 415: – In mathematics, process for extending a category 417:
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361: 294: 262: 194: 409: – Special case of colimit in category theory 379: 318: 280: 248: 434: 319:{\displaystyle \operatorname {GL} (\infty ,A)} 16:The direct limit of a direct system of groups 53:. Unsourced material may be challenged and 393:are the stable groups associated with the 117:Learn how and when to remove this message 184:forms a direct system, via inclusion of 149:. These are central objects of study in 424: – Construction in category theory 435: 380:{\displaystyle \operatorname {GL} (A)} 281:{\displaystyle \operatorname {GL} (A)} 332:computes the homotopy of the stable 51:adding citations to reliable sources 18: 13: 304: 188:in the upper left corner, such as 14: 464: 391:Stable homotopy groups of spheres 256:. The stable groups are denoted 23: 374: 368: 313: 301: 275: 269: 243: 225: 216: 213: 201: 1: 428: 355:) can be defined in terms of 7: 400: 175: 10: 469: 66:"Direct limit of groups" 381: 320: 282: 250: 155:stable homotopy theory 135:direct limit of groups 382: 321: 283: 251: 359: 292: 260: 192: 47:improve this article 448:Homological algebra 159:homological algebra 453:Algebraic topology 395:suspension functor 377: 316: 278: 246: 151:algebraic topology 127: 126: 119: 101: 460: 418: 386: 384: 383: 378: 338:orthogonal group 330:Bott periodicity 325: 323: 322: 317: 287: 285: 284: 279: 255: 253: 252: 247: 182:classical groups 122: 115: 111: 108: 102: 100: 59: 27: 19: 468: 467: 463: 462: 461: 459: 458: 457: 443:Homotopy theory 433: 432: 431: 416: 403: 360: 357: 356: 345:Whitehead group 293: 290: 289: 261: 258: 257: 193: 190: 189: 180:Each family of 178: 123: 112: 106: 103: 60: 58: 44: 28: 17: 12: 11: 5: 466: 456: 455: 450: 445: 430: 427: 426: 425: 419: 410: 402: 399: 376: 373: 370: 367: 364: 315: 312: 309: 306: 303: 300: 297: 277: 274: 271: 268: 265: 245: 242: 239: 236: 233: 230: 227: 224: 221: 218: 215: 212: 209: 206: 203: 200: 197: 177: 174: 171: 144: 125: 124: 31: 29: 22: 15: 9: 6: 4: 3: 2: 465: 454: 451: 449: 446: 444: 441: 440: 438: 423: 422:Inverse limit 420: 414: 413:Direct system 411: 408: 405: 404: 398: 396: 392: 388: 371: 365: 362: 354: 353:first K-group 350: 346: 341: 339: 335: 334:unitary group 331: 327: 310: 307: 298: 295: 272: 266: 263: 240: 237: 234: 231: 228: 222: 219: 210: 207: 204: 198: 195: 187: 183: 173: 169: 166: 164: 160: 156: 153:, especially 152: 148: 143:direct system 142: 140: 136: 132: 121: 118: 110: 99: 96: 92: 89: 85: 82: 78: 75: 71: 68: –  67: 63: 62:Find sources: 56: 52: 48: 42: 41: 37: 32:This article 30: 26: 21: 20: 407:Direct limit 389: 342: 328: 179: 167: 163:model theory 139:direct limit 134: 128: 113: 104: 94: 87: 80: 73: 61: 45:Please help 33: 336:and stable 131:mathematics 437:Categories 429:References 77:newspapers 366:⁡ 305:∞ 299:⁡ 267:⁡ 223:⁡ 217:→ 199:⁡ 107:June 2020 34:does not 401:See also 186:matrices 176:Examples 170:a priori 137:is the 91:scholar 55:removed 40:sources 147:groups 93:  86:  79:  72:  64:  351:(the 347:of a 141:of a 98:JSTOR 84:books 349:ring 343:The 157:and 133:, a 70:news 38:any 36:cite 288:or 165:. 145:of 129:In 49:by 439:: 397:. 387:. 363:GL 340:. 326:. 296:GL 264:GL 220:GL 196:GL 375:) 372:A 369:( 314:) 311:A 308:, 302:( 276:) 273:A 270:( 244:) 241:A 238:, 235:1 232:+ 229:n 226:( 214:) 211:A 208:, 205:n 202:( 120:) 114:( 109:) 105:( 95:· 88:· 81:· 74:· 57:. 43:.

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adding citations to reliable sources
removed
"Direct limit of groups"
news
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scholar
JSTOR
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mathematics
direct limit
groups
algebraic topology
stable homotopy theory
homological algebra
model theory
classical groups
matrices
Bott periodicity
unitary group
orthogonal group
Whitehead group
ring
first K-group
Stable homotopy groups of spheres
suspension functor

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