426:
252:
is metabelian. This is similar to the above example, as the elements are again affine maps. The translations of the plane form an abelian normal subgroup of the group, and the corresponding quotient is the
140:
180:
216:
467:
381:
460:
496:
453:
486:
90:
491:
231:
110:
101:
86:
29:
25:
153:
278:
192:
58:
377:
332:
300:
282:
433:
286:
261:
373:
322:
293:
249:
245:
41:
437:
82:
69:
65:
57:
Subgroups of metabelian groups are metabelian, as are images of metabelian groups over
48:
480:
33:
254:
223:
150:
is metabelian. Here the abelian normal subgroup is the group of pure translations
411:
357:
425:
187:
17:
183:
105:
331:
of order 24 is not metabelian, as its commutator subgroup is the non-abelian
277:
is metabelian. The same is true for any
Heisenberg group defined over a
400:
93:
is metabelian, as it has an abelian normal subgroup of index 2.
441:
195:
156:
113:
85:
is metabelian, as it has a cyclic normal subgroup of
68:. In fact, they are precisely the solvable groups of
210:
174:
134:
40:is metabelian if and only if there is an abelian
317:All groups of order less than 24 are metabelian.
461:
8:
468:
454:
194:
155:
112:
350:
410:Groupprops, The Group Properties Wiki
321:In contrast to this last example, the
230:elements, this metabelian group is of
7:
422:
420:
182:, and the abelian quotient group is
296:of class 3 or less are metabelian.
14:
285:3 × 3 matrices with entries in a
424:
370:A Course in the Theory of Groups
199:
160:
117:
1:
368:Robinson, Derek J.S. (1996),
135:{\displaystyle x\mapsto ax+b}
440:. You can help Knowledge by
175:{\displaystyle x\mapsto x+b}
211:{\displaystyle x\mapsto ax}
513:
419:
310:are metabelian (for prime
91:generalized dihedral group
36:. Equivalently, a group
89:2. More generally, any
497:Abstract algebra stubs
436:-related article is a
212:
176:
136:
64:Metabelian groups are
213:
177:
137:
487:Properties of groups
372:, Berlin, New York:
306:All groups of order
193:
154:
111:
59:group homomorphisms
30:commutator subgroup
208:
172:
132:
449:
448:
405:Metabelian Groups
383:978-0-387-94461-6
333:alternating group
301:lamplighter group
246:direct isometries
504:
470:
463:
456:
434:abstract algebra
428:
421:
412:Metabelian group
386:
360:
355:
294:nilpotent groups
287:commutative ring
283:upper-triangular
262:Heisenberg group
217:
215:
214:
209:
186:to the group of
181:
179:
178:
173:
141:
139:
138:
133:
22:metabelian group
512:
511:
507:
506:
505:
503:
502:
501:
492:Solvable groups
477:
476:
475:
474:
417:
401:Solvable groups
393:
384:
374:Springer-Verlag
367:
364:
363:
356:
352:
347:
340:
330:
323:symmetric group
272:
250:Euclidean plane
191:
190:
152:
151:
146:≠ 0) acting on
109:
108:
104:, the group of
78:
42:normal subgroup
12:
11:
5:
510:
508:
500:
499:
494:
489:
479:
478:
473:
472:
465:
458:
450:
447:
446:
429:
415:
414:
408:
392:
391:External links
389:
388:
387:
382:
362:
361:
349:
348:
346:
343:
338:
328:
319:
318:
315:
304:
303:is metabelian.
297:
290:
267:
258:
242:
207:
204:
201:
198:
171:
168:
165:
162:
159:
131:
128:
125:
122:
119:
116:
94:
83:dihedral group
77:
74:
70:derived length
49:quotient group
47:such that the
13:
10:
9:
6:
4:
3:
2:
509:
498:
495:
493:
490:
488:
485:
484:
482:
471:
466:
464:
459:
457:
452:
451:
445:
443:
439:
435:
430:
427:
423:
418:
413:
409:
406:
402:
398:
397:Ryan Wisnesky
395:
394:
390:
385:
379:
375:
371:
366:
365:
359:
354:
351:
344:
342:
337:
334:
327:
324:
316:
313:
309:
305:
302:
298:
295:
291:
288:
284:
280:
276:
271:
266:
263:
259:
256:
251:
247:
244:The group of
243:
240:
236:
233:
229:
225:
221:
205:
202:
196:
189:
185:
169:
166:
163:
157:
149:
145:
129:
126:
123:
120:
114:
107:
103:
99:
95:
92:
88:
84:
80:
79:
75:
73:
71:
67:
62:
60:
55:
53:
50:
46:
43:
39:
35:
31:
27:
23:
19:
442:expanding it
431:
416:
404:
403:(subsection
396:
369:
353:
335:
325:
320:
311:
307:
274:
269:
264:
255:circle group
238:
234:
227:
224:finite field
219:
147:
143:
97:
63:
56:
54:is abelian.
51:
44:
37:
21:
15:
260:The finite
188:homotheties
106:affine maps
72:at most 2.
18:mathematics
481:Categories
345:References
281:(group of
184:isomorphic
273:of order
200:↦
161:↦
118:↦
76:Examples
66:solvable
248:of the
142:(where
34:abelian
380:
28:whose
432:This
241:− 1).
232:order
226:with
222:is a
218:. If
102:field
100:is a
87:index
26:group
24:is a
438:stub
378:ISBN
299:The
292:All
279:ring
81:Any
20:, a
358:MSE
96:If
52:G/A
32:is
16:In
483::
399:,
376:,
341:.
314:).
289:).
268:3,
61:.
469:e
462:t
455:v
444:.
407:)
339:4
336:A
329:4
326:S
312:p
308:p
275:p
270:p
265:H
257:.
239:q
237:(
235:q
228:q
220:F
206:x
203:a
197:x
170:b
167:+
164:x
158:x
148:F
144:a
130:b
127:+
124:x
121:a
115:x
98:F
45:A
38:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.