328:, , , , , , , , , , , , , , , , , , , , , , , , , ] gap: -->
84:
74:
53:
22:
327:
List(SizesConjugacyClasses(ct),StringPP)])); , , , ,
218:
Order = 1, Size = 1 Order = 19, Size = 2643840 Order = 19, Size = 2643840 Order = 15, Size = 3348864 Order = 15, Size = 3348864 Order = 5, Size = 1674432 Order = 5, Size = 1674432 Order = 3, Size = 46512 Order = 10, Size = 5023296 Order = 10, Size = 5023296 Order = 2, Size = 26163 Order = 12, Size =
352:
The page in the ATLAS website says that elements of orders 5, 10, 15, 17, and 19 all fall into 2 power equivalent conjugacy classes. The 3 conjugacy classes of 9-elements also have what could be called power equivalency. Classes 3A and 3B are not power equivalent.
203:
g := Group(b11, b21);; Print("Size of group = ", Size(g), "\n\n");; c := ConjugacyClasses(g);; #f := ;; for e in do r := Representative(c);; #Add(f, ), 1]);; Print("Order = ", Order(r), ", Size = ", Size(c), "\n");; od; #f := SortedList(f); Print("\n");;
219:
4186080 Order = 6, Size = 2093040 Order = 4, Size = 523260 Order = 17, Size = 2954880 Order = 17, Size = 2954880 Order = 9, Size = 1860480 Order = 9, Size = 1860480 Order = 9, Size = 1860480 Order = 3, Size = 206720 Order = 8, Size = 6279120
389:
It is customary to include the fixed points (= orbits of length =1) when using the (permutation- or cycle-) shape of a permutation - thus the M24 example given is written 5^4.1^4 so that the degree ( = 24) may be deduced from the shape.
294:
For GAP, you can handle much larger groups using its character table library. The following gives all the sizes, orders, and power relations for the O'Nan group. To check that 8a and 8b are not power conjugate, use the last
368:
I agree, or at least so does GAP and the paper atlas. If we ever want the specific powers: I get that 19A = 19B^-1, 5A=5B^2, 9C=9B^2=9A^4, 15A=15B^2, 10A=10B^3, and 17A=17B^3, but 3A and 3B are different sizes.
140:
291:
by (a1, a1^g, a1^(g^2), ... )(a2, a2^g, ...) where each tuple has no duplicate entries. This is also called "cycle notation", and the "cycle structure" is the list of cycle sizes.
242:
When I said cycle structure I was referring to the structure of an element as a permutation in a well-known permutation representation. For example, in a representation of M
441:
130:
436:
358:
261:
106:
337:
If you like to program, GAP can be very helpful. Otherwise the online ATLAS and the paper ATLAS have approximately the same information.
394:
354:
257:
172:
97:
58:
33:
208:
190:
165:
Further, McKay & Wales gave the groups their names so that their index was the order of their Schur multiplier.
21:
398:
176:
374:
342:
271:
As near as I can tell, wikipedia has no article on "cycle structure". The cycle structure of an element
197:
a lot faster, and with a lot less memory than a program I wrote that generates all elements in a group.
39:
83:
168:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
73:
52:
418:
370:
338:
275:
of a permutation group is the multi-set of sizes of orbits of the cyclic group generated by
207:
my computer didn't run out of memory. Not sure how to easily check for power equivalence in
212:
194:
412:
Reduced large white space at 1024x768 by moving the infobox below the long <math: -->
430:
317:
SizesConjugacyClasses(ct); # See which classes are 13th powers of each other gap: -->
228:
287:
only has finite orbits, which takes orbit representatives a1, a2, ... and specifies
246:
of degree 24, an element of order 5 is a product of 4 disjoint 5-cycles, written 5.
414:
102:
79:
239:, but it lists orders of centralizers instead of sized of conjugacy classes.
314:
LoadPackage("ctbllib");; # Get the O'Nan group as a character table gap: -->
222:
422:
402:
378:
362:
346:
265:
230:
413:
tags. Better to shrink the huge infobox itself, IMHO. Discuss. --
249:
On the ATLAS website the smallest permutation representation of J
235:
Robert A, Wilson's ATLAS website lists the conjugacy classes of J
15:
162:
The correction to the Higman - Mckay reference is omitted.
253:
has degree 6156. I am not inclined to bother with it.
215:
is. Maybe it's easy for someone with GAP experience.
211:
or how to print factors, and don't even know what a
101:, a collaborative effort to improve the coverage of
324:PrintArray(TransposedMat([ClassNames(ct), : -->
279:. This is related to the cycle decomposition of
323:ClassNames(ct){x}); , , , , , , ] gap: -->
8:
256:The ATLAS listing shows power equivalence.
19:
319:Cycles(PermList(PowerMap(ct,13)), ), : -->
298:
47:
316:OrdersClassRepresentatives(ct); gap: -->
49:
200:Using the generators and this script:
325:OrdersClassRepresentatives(ct), : -->
315:ct := CharacterTable("ON");; gap: -->
7:
95:This article is within the scope of
38:It is of interest to the following
14:
442:Low-priority mathematics articles
115:Knowledge:WikiProject Mathematics
437:Start-Class mathematics articles
326:SizesConjugacyClasses(ct), : -->
185:Number of elements of each order
118:Template:WikiProject Mathematics
82:
72:
51:
20:
135:This article has been rated as
403:11:02, 22 September 2009 (UTC)
1:
355:Scott Tillinghast, Houston TX
329:PrintArray(TransposedMat());
258:Scott Tillinghast, Houston TX
109:and see a list of open tasks.
458:
423:10:53, 27 March 2010 (UTC)
134:
67:
46:
379:14:38, 9 July 2008 (UTC)
363:14:16, 9 July 2008 (UTC)
347:15:33, 8 July 2008 (UTC)
266:00:06, 9 July 2008 (UTC)
231:20:57, 5 July 2008 (UTC)
158:Requests for corrections
141:project's priority scale
193:, and it seems to find
98:WikiProject Mathematics
28:This article is rated
385:Shape of permutations
318:List( Filtered( : -->
307:O'Nan example in GAP
302:O'Nan example in GAP
179:) 2008-06-06T09:32:47
121:mathematics articles
90:Mathematics portal
34:content assessment
334:
333:
195:conjugacy classes
189:Tried installing
180:
171:comment added by
155:
154:
151:
150:
147:
146:
449:
299:
166:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
457:
456:
452:
451:
450:
448:
447:
446:
427:
426:
410:
408:Page appearance
387:
330:
252:
245:
238:
220:
213:cycle structure
205:
187:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
455:
453:
445:
444:
439:
429:
428:
409:
406:
386:
383:
382:
381:
350:
349:
332:
331:
321:Length(x): -->
312:
309:
308:
304:
303:
297:
296:
292:
250:
243:
236:
217:
202:
186:
183:
159:
156:
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:
454:
443:
440:
438:
435:
434:
432:
425:
424:
420:
416:
407:
405:
404:
400:
396:
395:66.130.86.141
391:
384:
380:
376:
372:
367:
366:
365:
364:
360:
356:
348:
344:
340:
336:
335:
311:
310:
306:
305:
301:
300:
293:
290:
286:
282:
278:
274:
270:
269:
268:
267:
263:
259:
254:
247:
240:
233:
232:
229:
227:
226:
216:
214:
210:
201:
198:
196:
192:
184:
182:
178:
174:
170:
163:
157:
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:
27:
23:
18:
17:
411:
392:
388:
351:
288:
284:
280:
276:
272:
255:
248:
241:
234:
224:
221:
206:
199:
188:
173:130.54.16.90
164:
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
393:John McKay
371:JackSchmidt
339:JackSchmidt
322:1), x-: -->
167:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
431:Categories
295:commands.
313:gap: -->
169:unsigned
320:x-: -->
139:on the
415:Lexein
36:scale.
283:when
223:Κσυπ
419:talk
399:talk
375:talk
359:talk
343:talk
262:talk
177:talk
225:Cyp
209:GAP
191:GAP
131:Low
433::
421:)
401:)
377:)
361:)
345:)
264:)
244:24
181:.
417:(
397:(
373:(
357:(
341:(
289:g
285:g
281:g
277:g
273:g
260:(
251:3
237:3
175:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
