321:(3) The use of the term orbits here and in the article is at best obscure and certainly not conventional in the sense of popular usage. The mathematical structure of an orbit and how an orbit is generated by a group action are in no way similar to the structure and generation implied by the usage of the term here. This usage should not be promoted in general and it should not be repeated here as it only leads to confusion (despite the fact that at least one mathematics book, as cited above, has been published stipulating this usage). The term orbits as used in the article should be replaced with a descriptive phrase such as collections of cycle elements (referring to the elements used in the denotation of a cyclic permutation).
318:(2) The use of the term orbits in this phrase is circular in its use as a means of defining the word orbit. Also, it never describes how a cyclic group (presumed to mean not a group of, but a simple collection of cyclic permutations) is related to its orbits. Here, the use of the term orbit is probably intended to refer to the orbit for a cyclic permutation, which in turn is intended to mean the collection of elements listed in the denotation of the cyclic permutation when the permutation is denoted in cycle notation.
84:
74:
53:
22:
315:(1) What is cyclic group? How is it generated? It may be that the intended meaning is the reverse. It is known that any permutation is generated by a product of cyclic permutations. The collection of cyclic permutations that generate a given permutation, however does not necessarily form a group. The phrase would then become "orbits of the collection of cyclic permutations which generate that permutation."
255:
The orbits of a single permutation are the same as the orbits of the cyclic group generated by that permutation. The orbits of (1,2,3)(4,5)(6)(7) = (1,2,3)(4,5) = on {1,2,3,4,5,6,7} are {1,2,3}, {4,5}, {6}, and {7}. If you omit 1-cycles, then you must be explicit about the domain being acted on.
161:
Seems O(n) to me: you start with the first element, compute the output of the permutation, and use that as input. If the output is the first element again, you start over with another unused element, the used elements are one cycle. Each element of the permutation will be used exactly once, and the
300:
Do not agree. This usage and definition of the term orbit should not be included on the
Knowledge page defining orbit. It is unconventional and completely different from the common and proper usage of the
277:
Thanks for answering Jack! Perhaps we should include this definition in that of orbit? In most places, this is not mentioned, but I finally came across a book by
Gerhard Michler which gave this definition.
442:
which is what the definition is trying to say. The author of the article probably thought that the concept of a cyclic group generated by an element would be understood by readers of this page.
140:
438:) is {1,2,3}, the orbit of 4 is {4,5}, the orbit of 6 is {6} and so on. Notice how the elements in an orbit are precisely the elements in a cycle of the original permutation
464:
This article omits motivation and barely has examples. At the very least, it should explain how group multiplication works for cycles. Moly 15:58, 23 October 2012 (UTC)
229:
209:
497:
502:
130:
106:
162:
cycles will be disjoint (this is the exact technique used to prove the theorem). I don't think it'd be possible to do in a better time.
97:
58:
328:
352:
Jack was quite correct and the use of the term orbit in this article is quite standard and in agreement with the article
33:
480:
21:
261:
39:
83:
170:
468:
332:
324:
188:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
447:
279:
241:
166:
89:
356:. Perhaps Jack's explanation should be fleshed out and given in this article. Given any element
73:
52:
231:
acts upon, but not of the group element itself. Can someone please clarify in what context is
257:
476:
214:
194:
491:
443:
157:
What is the algorithmic complexity of decomposing a permutation into those cycles?
239:
permutation is being concerned, then should any element not have only one orbit?
102:
79:
472:
312:"orbits of the cyclic group generated by that permutation" is not clear.
451:
336:
287:
265:
249:
174:
353:
389:, etc. when the group is finite and written multiplicatively. If
15:
401:), then this subgroup acts naturally on the set {1,2,...,
217:
197:
101:, a collaborative effort to improve the coverage of
223:
203:
8:
191:defines the orbit of an element of the set
433:. The orbit of 1 under this subgroup (<
322:
47:
216:
196:
32:does not require a rating on Knowledge's
49:
95:This redirect is within the scope of
19:
7:
498:Redirect-Class mathematics articles
38:It is of interest to the following
14:
503:Low-priority mathematics articles
115:Knowledge:WikiProject Mathematics
373:) is the subgroup consisting of
364:, the cyclic group generated by
183:permutation as a group of cycles
135:This redirect has been rated as
118:Template:WikiProject Mathematics
82:
72:
51:
20:
1:
452:23:52, 30 December 2014 (UTC)
337:21:44, 30 December 2014 (UTC)
235:being used here? Also, since
109:and see a list of open tasks.
175:13:32, 1 January 2011 (UTC)
165:You should sign your name.
519:
409:=(123)(45)(6)(7), we have
429:=(1)(2)(3)(4)(5)(6)(7) =
405:}. Using Jack's example,
393:is the permutation group
288:00:36, 3 March 2011 (UTC)
266:15:20, 2 March 2011 (UTC)
250:06:39, 2 March 2011 (UTC)
134:
67:
46:
368:(usually denoted by <
141:project's priority scale
98:WikiProject Mathematics
417:=(1)(2)(3)(45)(6)(7),
225:
205:
187:The Knowledge page on
425:=(132)(45)(6)(7) and
413:= (132)(4)(5)(6)(7),
226:
206:
421:=(123)(4)(5)(6)(7),
215:
195:
121:mathematics articles
221:
201:
90:Mathematics portal
34:content assessment
485:
471:comment added by
339:
327:comment added by
286:
248:
224:{\displaystyle G}
211:on which a group
204:{\displaystyle S}
155:
154:
151:
150:
147:
146:
510:
484:
465:
444:Bill Cherowitzo
285:
282:
247:
244:
230:
228:
227:
222:
210:
208:
207:
202:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
25:
24:
16:
518:
517:
513:
512:
511:
509:
508:
507:
488:
487:
466:
280:
242:
213:
212:
193:
192:
185:
120:
117:
114:
111:
110:
88:
81:
61:
12:
11:
5:
516:
514:
506:
505:
500:
490:
489:
463:
461:
460:
459:
458:
457:
456:
455:
454:
343:
342:
341:
340:
319:
316:
313:
307:
306:
305:
304:
303:
302:
293:
292:
291:
290:
271:
269:
268:
220:
200:
184:
181:
180:
179:
178:
177:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
515:
504:
501:
499:
496:
495:
493:
486:
482:
478:
474:
470:
453:
449:
445:
441:
436:
432:
428:
424:
420:
416:
412:
408:
404:
400:
396:
392:
388:
384:
380:
376:
371:
367:
363:
360:of any group
359:
355:
351:
350:
349:
348:
347:
346:
345:
344:
338:
334:
330:
326:
320:
317:
314:
311:
310:
309:
308:
299:
298:
297:
296:
295:
294:
289:
284:
283:
281:Pratik.Mallya
276:
275:
274:
273:
272:
267:
263:
259:
254:
253:
252:
251:
246:
245:
243:Pratik.Mallya
238:
234:
218:
198:
190:
182:
176:
172:
168:
164:
163:
160:
159:
158:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
31:
27:
23:
18:
17:
467:— Preceding
462:
439:
434:
430:
426:
422:
418:
414:
410:
406:
402:
398:
394:
390:
386:
382:
378:
374:
369:
365:
361:
357:
323:— Preceding
278:
270:
240:
236:
232:
186:
156:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
29:
258:JackSchmidt
112:Mathematics
103:mathematics
59:Mathematics
492:Categories
329:66.194.2.9
481:contribs
469:unsigned
325:unsigned
30:redirect
139:on the
189:orbits
167:Zyszys
36:scale.
437:: -->
372:: -->
354:orbit
301:term.
233:orbit
28:This
477:talk
473:Moly
448:talk
333:talk
262:talk
171:talk
237:one
131:Low
494::
483:)
479:•
450:)
385:,
381:,
377:,
335:)
264:)
173:)
475:(
446:(
440:g
435:g
431:e
427:g
423:g
419:g
415:g
411:g
407:g
403:n
399:n
397:(
395:S
391:G
387:g
383:g
379:g
375:e
370:g
366:g
362:G
358:g
331:(
260:(
219:G
199:S
169:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.