1594:
6936:
52:
5942:
2622:
1391:
3365:
4642:
2153:
4379:
5761:
1896:
2440:
1589:{\displaystyle \mathbb {Q} \subseteq \mathbb {Q} ({\sqrt {2}})\subseteq \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\subseteq \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5}\right)\subseteq \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5},a\right)}
6025:
6099:
6349:
0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, ... (sequence
1682:
3197:
2303:
4507:
2011:
5208:
1003:
2963:
3892:
3638:
3511:
4262:
5122:
4754:
3772:
The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series is formed by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.
1971:
1757:
5937:{\displaystyle B=\left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&0&*\end{bmatrix}}\right\},{\text{ }}U_{1}=\left\{{\begin{bmatrix}1&*&*\\0&1&*\\0&0&1\end{bmatrix}}\right\}}
5444:
6786:
stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a
6209:
5500:
5004:
2770:
4061:
2796:
is the identity permutation. All of the defining group actions change a single extension while keeping all of the other extensions fixed. For example, an element of this group is the group action
2413:
1800:
3073:
3727:
guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to
1380:
2713:
6368:
60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, ... (sequence
2617:{\displaystyle \mathrm {Aut} \left(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5},a\right)/\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5}\right)\right)\cong \mathbb {Z} /5}
1129:
3151:
5950:
4420:
6153:
6033:
4928:
3406:
5547:
3925:
4089:
1071:
4900:
4445:
2837:
1609:
1326:
1235:
497:
472:
435:
5310:
1196:
1156:
6304:
6258:
3360:{\displaystyle (\mathbb {C} _{5}\rtimes _{\varphi }\mathbb {C} _{4})\times (\mathbb {C} _{2}\times \mathbb {C} _{2}),\ \mathrm {where} \ \varphi _{h}(j)=hjh^{-1}=j^{2}}
3192:
4704:
4004:
2004:
1793:
6875:
5753:
5712:
5662:
5632:
5044:
4878:
4841:
4675:
916:
7094:
5240:
4637:{\displaystyle {\begin{bmatrix}a&b\\0&c\end{bmatrix}}\cdot {\begin{bmatrix}1&d\\0&1\end{bmatrix}}={\begin{bmatrix}a&ad+b\\0&c\end{bmatrix}}}
4136:
states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.
1263:
2148:{\displaystyle \mathrm {Aut} \left(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5}\right)/\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\right)\cong \mathbb {Z} /4}
1296:
4806:
4780:
4499:
4473:
2158:
799:
3948:
6278:
6232:
5682:
5602:
5578:
5260:
5150:
5064:
4948:
3979:
2794:
2433:
5155:
924:
6609:. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.
2842:
6893:
3817:
3576:
3445:
4374:{\displaystyle B=\left\{{\begin{bmatrix}*&*\\0&*\end{bmatrix}}\right\}{\text{, }}U=\left\{{\begin{bmatrix}1&*\\0&1\end{bmatrix}}\right\}}
6375:
6357:
5069:
4709:
357:
7071:
1901:
1687:
307:
792:
302:
7019:
6161:
5452:
4956:
1760:
5664:
the groups of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup
2718:
6953:
1891:{\displaystyle \mathrm {Aut} \left(\mathbb {Q({\sqrt {2}},{\sqrt {3}})} \right/\mathbb {Q({\sqrt {2}})} )\cong \mathbb {Z} /2}
7000:
6957:
718:
5604:, and is a maximal possible subgroup with these properties (note the first two are topological properties). For example, in
6972:
5315:
3735:. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group
2308:
7114:
6659:
2977:
785:
4166:
3724:
1334:
6722:
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:
2627:
6979:
6774:. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.
6594:
402:
216:
1076:
4133:
880:
6986:
4238:
7109:
6750:
6709:
6606:
6487:
3372:
600:
334:
211:
99:
4012:
3079:
6020:{\displaystyle B/U_{1}\cong \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }}
7039:
6946:
6968:
6094:{\displaystyle U\rtimes (\mathbb {F} ^{\times }\times \mathbb {F} ^{\times }\times \mathbb {F} ^{\times })}
5558:
4384:
4169:
states that every other composition series is equivalent to that one), giving factor groups isomorphic to
750:
540:
6112:
624:
6712:, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group
6618:
4905:
3382:
6632:
5508:
3716:
1677:{\displaystyle \mathrm {Aut} \left(\mathbb {Q({\sqrt {2}})} \right/\mathbb {Q} )\cong \mathbb {Z} /2}
564:
552:
170:
104:
3900:
6738:
6704:
normal series whose factors are all cyclic. Since a normal series has finite length by definition,
6683:
6433:
3732:
3644:
1014:
865:
829:
139:
34:
6459:
The previous properties can be expanded into the following "three for the price of two" property:
4883:
4428:
2799:
1309:
1201:
480:
455:
418:
6534:
6415:
6322:
5265:
3705:
3368:
1600:
1161:
1134:
124:
96:
6283:
6237:
3158:
7067:
6834:
6771:
4680:
4117:
3744:
695:
529:
372:
266:
6821:
A finite group is p-solvable for some prime p if every factor in the composition series is a
1976:
1765:
7059:
6993:
6742:
5728:
5687:
5637:
5607:
5012:
4846:
4811:
4650:
3808:
3425:
2298:{\displaystyle h^{n}\left(e^{2im\pi /5}\right)=e^{2(n+1)mi\pi /5},\ 0\leq n\leq 3,\ h^{4}=1}
886:
845:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
134:
7052:
5213:
1241:
7048:
7034:
6734:
6502:
4242:
4100:
3958:
3782:
3656:
3527:
1272:
1266:
837:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
5203:{\displaystyle \mathbb {F} \rtimes (\mathbb {F} ^{\times }\times \mathbb {F} ^{\times })}
4785:
4759:
4478:
4450:
4069:
3984:
3930:
6825:
or has order prime to p. A finite group is solvable iff it is p-solvable for every p.
6586:
6562:
6263:
6217:
5667:
5587:
5581:
5563:
5245:
5135:
5049:
4933:
3964:
3704:
For finite groups, an equivalent definition is that a solvable group is a group with a
3675:
3567:
2779:
2418:
998:{\displaystyle F=F_{0}\subseteq F_{1}\subseteq F_{2}\subseteq \cdots \subseteq F_{m}=K}
872:
841:
771:
707:
397:
377:
314:
279:
190:
175:
160:
114:
91:
2958:{\displaystyle f^{a}g^{b}h^{n}j^{l},\ 0\leq a,b\leq 1,\ 0\leq n\leq 3,\ 0\leq l\leq 4}
7103:
6730:
3719:. This is equivalent because a finite group has finite composition length, and every
3433:
2969:
857:
833:
690:
612:
446:
319:
185:
3887:{\displaystyle 1\to \mathbb {Z} /2\to Q\to \mathbb {Z} /2\times \mathbb {Z} /2\to 1}
3633:{\displaystyle G\triangleright G^{(1)}\triangleright G^{(2)}\triangleright \cdots ,}
3506:{\displaystyle 1=G_{0}\triangleleft G_{1}\triangleleft \cdots \triangleleft G_{k}=G}
6898:
6880:
6789:
6726:
6647:
6628:
6598:
3720:
3713:
3709:
3429:
3376:
876:
817:
545:
244:
233:
180:
155:
150:
109:
80:
43:
6910:
6855:
6935:
6809:, and it has been shown that every ordinal is the derived length of some group (
6705:
6315:
813:
6782:
A solvable group is one whose derived series reaches the trivial subgroup at a
6602:
6422:
4230:
712:
440:
17:
5117:{\displaystyle B/U\cong \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }}
4749:{\displaystyle \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }\subset B}
533:
4501:, multiplying them together, and figuring out what structure this gives. So
51:
4091:. In fact, all solvable groups can be formed from such group extensions.
3437:
869:
70:
7066:, Graduate Texts in Mathematics, vol. 148 (4 ed.), Springer,
6822:
6662:
6326:
4190:
is not abelian. Generalizing this argument, coupled with the fact that
3793:
3786:
3740:
1966:{\displaystyle g\left(\pm {\sqrt {3}}\right)=\mp {\sqrt {3}},\ g^{2}=1}
1752:{\displaystyle f\left(\pm {\sqrt {2}}\right)=\mp {\sqrt {2}},\ f^{2}=1}
861:
412:
326:
7037:(1949), "Generalized nilpotent algebras and their associated groups",
6793:, and every solvable group is a hypoabelian group. The first ordinal
6325:
of two cyclic groups, in particular solvable. Such groups are called
1599:
giving a solvable group of Galois extensions containing the following
5584:
is defined as a subgroup which is closed, connected, and solvable in
6719:
is an example of a finite solvable group that is not supersolvable.
6708:
groups are not supersolvable. In fact, all supersolvable groups are
6770:
if it has a solvable subgroup of finite index. This is similar to
4241:). This property is also used in complexity theory in the proof of
7086:
6597:
of the variety of groups, as they are closed under the taking of
3751:
itself, it has no composition series, but the normal series {0,
3647:
of the previous one, eventually reaches the trivial subgroup of
3961:
form the prototypical examples of solvable groups. That is, if
6929:
6204:{\displaystyle {\begin{bmatrix}T&0\\0&S\end{bmatrix}}}
5495:{\displaystyle {\begin{bmatrix}a&b\\0&c\end{bmatrix}}}
4999:{\displaystyle {\begin{bmatrix}1&d\\0&1\end{bmatrix}}}
3651:. These two definitions are equivalent, since for every group
6155:
the Borel subgroup can be represented by matrices of the form
7089:
6370:
6352:
6105:
Borel subgroup in product of simple linear algebraic groups
1847:
1646:
7090:
sequence A056866 (Orders of non-solvable groups)
4099:
A small example of a solvable, non-nilpotent group is the
2765:{\displaystyle x^{5}-\left({\sqrt {2}}+{\sqrt {3}}\right)}
1865:
1855:
1843:
1823:
1642:
1632:
1385:
gives a solvable group. It has associated field extensions
27:
Group with subnormal series where all factors are abelian
6805:
is called the (transfinite) derived length of the group
5132:
Notice that this description gives the decomposition of
4151:
is not solvable — it has a composition series {E,
4225:> 4. This is a key step in the proof that for every
4109:. In fact, as the smallest simple non-abelian group is
6170:
5872:
5780:
5461:
4965:
4594:
4555:
4516:
4336:
4281:
6286:
6266:
6240:
6220:
6164:
6115:
6036:
5953:
5764:
5731:
5690:
5670:
5640:
5610:
5590:
5566:
5511:
5455:
5439:{\displaystyle (a,c)(b+b')=(a,c)(b)+(a,c)(b')=ab+ab'}
5318:
5268:
5248:
5216:
5158:
5138:
5072:
5052:
5015:
4959:
4936:
4908:
4886:
4849:
4814:
4788:
4762:
4712:
4683:
4653:
4510:
4481:
4453:
4431:
4387:
4265:
4199:
is a normal, maximal, non-abelian simple subgroup of
4072:
4015:
3987:
3967:
3933:
3903:
3820:
3579:
3448:
3385:
3200:
3161:
3082:
2980:
2845:
2802:
2782:
2721:
2630:
2443:
2421:
2408:{\displaystyle x^{4}+x^{3}+x^{2}+x+1=(x^{5}-1)/(x-1)}
2311:
2161:
2014:
1979:
1904:
1803:
1768:
1690:
1612:
1394:
1337:
1312:
1275:
1244:
1204:
1164:
1137:
1079:
1017:
927:
889:
483:
458:
421:
6387:
Solvability is closed under a number of operations.
3068:{\displaystyle hj(a)=h(e^{2i\pi /5}a)=e^{4i\pi /5}a}
1306:
For example, the smallest Galois field extension of
6960:. Unsourced material may be challenged and removed.
2839:. A general element in the group can be written as
840:. Equivalently, a solvable group is a group whose
6298:
6272:
6252:
6226:
6203:
6147:
6093:
6019:
5936:
5747:
5706:
5676:
5656:
5626:
5596:
5572:
5541:
5494:
5438:
5304:
5254:
5234:
5202:
5144:
5116:
5058:
5038:
4998:
4942:
4922:
4894:
4872:
4835:
4800:
4774:
4748:
4698:
4669:
4636:
4493:
4467:
4439:
4414:
4373:
4083:
4055:
3998:
3973:
3942:
3919:
3886:
3632:
3505:
3400:
3359:
3186:
3145:
3067:
2957:
2831:
2788:
2764:
2707:
2616:
2427:
2407:
2297:
2147:
1998:
1965:
1890:
1787:
1751:
1676:
1588:
1375:{\displaystyle a={\sqrt{{\sqrt {2}}+{\sqrt {3}}}}}
1374:
1320:
1290:
1257:
1229:
1190:
1150:
1123:
1065:
997:
910:
491:
466:
429:
2708:{\displaystyle j^{l}(a)=e^{2li\pi /5}a,\ j^{5}=1}
3811:is a solvable group given by the group extension
2968:It is worthwhile to note that this group is not
1124:{\displaystyle \alpha _{i}^{m_{i}}\in F_{i-1}}
860:and the proof of the general unsolvability of
4475:can be found by taking arbitrary elements in
879:is solvable (note this theorem holds only in
856:Historically, the word "solvable" arose from
793:
8:
4756:is a subgroup (which are the matrices where
3755:}, with its only factor group isomorphic to
3723:abelian group is cyclic of prime order. The
2415:containing the 5th roots of unity excluding
6688:As a strengthening of solvability, a group
4124:group with order less than 60 is solvable.
883:0). This means associated to a polynomial
800:
786:
238:
64:
29:
7020:Learn how and when to remove this message
6285:
6265:
6239:
6219:
6165:
6163:
6139:
6123:
6114:
6082:
6078:
6077:
6067:
6063:
6062:
6052:
6048:
6047:
6035:
6011:
6007:
6006:
5996:
5992:
5991:
5981:
5977:
5976:
5966:
5957:
5952:
5867:
5854:
5845:
5775:
5763:
5739:
5730:
5698:
5689:
5669:
5648:
5639:
5618:
5609:
5589:
5565:
5510:
5456:
5454:
5317:
5267:
5247:
5215:
5191:
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5186:
5176:
5172:
5171:
5160:
5159:
5157:
5137:
5108:
5104:
5103:
5093:
5089:
5088:
5076:
5071:
5051:
5028:
5014:
4960:
4958:
4935:
4916:
4915:
4907:
4888:
4887:
4885:
4862:
4848:
4813:
4787:
4761:
4734:
4730:
4729:
4719:
4715:
4714:
4711:
4682:
4661:
4652:
4589:
4550:
4511:
4509:
4480:
4457:
4452:
4433:
4432:
4430:
4405:
4404:
4395:
4386:
4331:
4316:
4276:
4264:
4071:
4014:
3986:
3966:
3932:
3909:
3905:
3904:
3902:
3870:
3866:
3865:
3854:
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3849:
3832:
3828:
3827:
3819:
3609:
3590:
3578:
3491:
3472:
3459:
3447:
3392:
3388:
3387:
3384:
3351:
3335:
3307:
3283:
3268:
3264:
3263:
3253:
3249:
3248:
3232:
3228:
3227:
3220:
3210:
3206:
3205:
3199:
3178:
3160:
3130:
3120:
3081:
3052:
3042:
3019:
3009:
2979:
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2870:
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2844:
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2813:
2801:
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2635:
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2497:
2478:
2468:
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2370:
2342:
2329:
2316:
2310:
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2242:
2214:
2193:
2180:
2166:
2160:
2137:
2133:
2132:
2114:
2104:
2097:
2096:
2091:
2077:
2067:
2049:
2039:
2032:
2031:
2015:
2013:
1984:
1978:
1951:
1934:
1916:
1903:
1880:
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1526:
1519:
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1473:
1463:
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1411:
1404:
1403:
1396:
1395:
1393:
1365:
1357:
1347:
1344:
1336:
1314:
1313:
1311:
1274:
1249:
1243:
1215:
1203:
1174:
1169:
1163:
1142:
1136:
1109:
1094:
1089:
1084:
1078:
1054:
1035:
1022:
1016:
983:
964:
951:
938:
926:
888:
485:
484:
482:
460:
459:
457:
423:
422:
420:
6541:It is also closed under wreath product:
4095:Non-abelian group which is non-nilpotent
4056:{\displaystyle 1\to G\to G''\to G'\to 1}
3146:{\displaystyle jh(a)=j(a)=e^{2i\pi /5}a}
7064:An Introduction to the Theory of Groups
6846:
4006:are solvable groups, then any extension
356:
122:
32:
7095:Solvable groups as iterated extensions
6810:
6337:Numbers of solvable groups with order
3759:, proves that it is in fact solvable.
358:Classification of finite simple groups
3194:. The solvable group is isometric to
7:
6958:adding citations to reliable sources
6027:, hence the Borel group has the form
4415:{\displaystyle GL_{2}(\mathbb {F} )}
4237:which are not solvable by radicals (
3785:are solvable. In particular, finite
3681:includes the commutator subgroup of
918:there is a tower of field extensions
816:, more specifically in the field of
6148:{\displaystyle GL_{n}\times GL_{m}}
4880:, which is an arbitrary element in
6623:Burnside's theorem states that if
6432:is solvable; equivalently (by the
6364:Orders of non-solvable groups are
5046:, we can get a diagonal matrix in
4930:. Since we can take any matrix in
4647:Note the determinant condition on
3296:
3293:
3290:
3287:
3284:
2451:
2448:
2445:
2022:
2019:
2016:
1811:
1808:
1805:
1620:
1617:
1614:
25:
4923:{\displaystyle b\in \mathbb {F} }
6934:
6463:is solvable if and only if both
5066:. This shows the quotient group
3401:{\displaystyle \mathbb {C} _{4}}
50:
6945:needs additional citations for
5542:{\displaystyle (b)\times (a,c)}
3570:, the descending normal series
6088:
6043:
5536:
5524:
5518:
5512:
5410:
5399:
5396:
5384:
5378:
5372:
5369:
5357:
5351:
5334:
5331:
5319:
5290:
5284:
5281:
5269:
5229:
5217:
5197:
5167:
4409:
4401:
4047:
4036:
4025:
4019:
3920:{\displaystyle \mathbb {Z} /2}
3878:
3846:
3840:
3824:
3731:th roots (radicals) over some
3616:
3610:
3597:
3591:
3319:
3313:
3274:
3244:
3238:
3201:
3110:
3104:
3095:
3089:
3032:
3002:
2993:
2987:
2647:
2641:
2558:
2538:
2485:
2465:
2402:
2390:
2382:
2363:
2230:
2218:
2121:
2101:
2056:
2036:
1869:
1655:
1543:
1523:
1480:
1460:
1449:
1429:
1418:
1408:
1285:
1279:
1158:is a solution to the equation
1060:
1047:
905:
899:
719:Infinite dimensional Lie group
1:
4950:and multiply it by the matrix
4120:of degree 5) it follows that
3927:is the subgroup generated by
1066:{\displaystyle F_{i}=F_{i-1}}
832:that can be constructed from
6501:Solvability is closed under
6414:is solvable, and there is a
6260:upper triangular matrix and
5446:. Also, a matrix of the form
4895:{\displaystyle \mathbb {F} }
4440:{\displaystyle \mathbb {F} }
3792:are solvable, as all finite
3643:where every subgroup is the
2965:for a total of 80 elements.
2832:{\displaystyle fgh^{3}j^{4}}
1321:{\displaystyle \mathbb {Q} }
1230:{\displaystyle a\in F_{i-1}}
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
5505:corresponds to the element
5305:{\displaystyle (a,c)(b)=ab}
4447:. Then, the group quotient
1191:{\displaystyle x^{m_{i}}-a}
1151:{\displaystyle \alpha _{i}}
864:equations. Specifically, a
217:List of group theory topics
7131:
6681:
6616:
4128:Finite groups of odd order
3432:(quotient groups) are all
3408:is not a normal subgroup.
3379:. In the solvable group,
6758:Virtually solvable groups
6585:, the solvable groups of
6581:For any positive integer
6521:are solvable, then so is
6434:first isomorphism theorem
6306:upper triangular matrix.
6299:{\displaystyle m\times m}
6253:{\displaystyle n\times n}
4066:defines a solvable group
3708:all of whose factors are
3555:is an abelian group, for
3187:{\displaystyle jh=hj^{3}}
2715:, and minimal polynomial
2305:, and minimal polynomial
1973:, and minimal polynomial
6751:finitely generated group
6444:is a normal subgroup of
4699:{\displaystyle ac\neq 0}
3566:Or equivalently, if its
3436:, that is, if there are
3155:In fact, in this group,
335:Elementary abelian group
212:Glossary of group theory
6612:
6314:Any finite group whose
5755:there are the subgroups
5714:, is a Borel subgroup.
1999:{\displaystyle x^{2}-3}
1788:{\displaystyle x^{2}-2}
6300:
6274:
6254:
6228:
6212:
6205:
6149:
6102:
6095:
6021:
5945:
5938:
5749:
5748:{\displaystyle GL_{3}}
5708:
5707:{\displaystyle GL_{2}}
5678:
5658:
5657:{\displaystyle SL_{n}}
5628:
5627:{\displaystyle GL_{n}}
5598:
5574:
5559:linear algebraic group
5543:
5503:
5496:
5440:
5306:
5256:
5236:
5204:
5146:
5118:
5060:
5040:
5039:{\displaystyle d=-b/a}
5007:
5000:
4944:
4924:
4896:
4874:
4873:{\displaystyle d=-b/a}
4837:
4836:{\displaystyle ad+b=0}
4808:, the linear equation
4802:
4776:
4750:
4700:
4671:
4670:{\displaystyle GL_{2}}
4645:
4638:
4495:
4469:
4441:
4423:
4416:
4375:
4256:Consider the subgroups
4085:
4064:
4057:
4000:
3975:
3944:
3921:
3895:
3888:
3697:of the solvable group
3634:
3507:
3402:
3361:
3188:
3147:
3069:
2972:itself. For example:
2959:
2833:
2790:
2766:
2709:
2618:
2429:
2409:
2299:
2149:
2000:
1967:
1892:
1789:
1753:
1678:
1597:
1590:
1383:
1376:
1328:containing the element
1322:
1292:
1259:
1231:
1192:
1152:
1125:
1067:
1006:
999:
912:
911:{\displaystyle f\in F}
751:Linear algebraic group
493:
468:
431:
6301:
6275:
6255:
6229:
6206:
6157:
6150:
6109:In the product group
6096:
6029:
6022:
5939:
5757:
5750:
5709:
5679:
5659:
5629:
5599:
5575:
5544:
5497:
5448:
5441:
5307:
5257:
5237:
5235:{\displaystyle (a,c)}
5205:
5147:
5119:
5061:
5041:
5001:
4952:
4945:
4925:
4897:
4875:
4838:
4803:
4777:
4751:
4701:
4672:
4639:
4503:
4496:
4470:
4442:
4417:
4376:
4258:
4167:Jordan–Hölder theorem
4134:Feit–Thompson theorem
4086:
4058:
4008:
4001:
3976:
3945:
3922:
3889:
3813:
3725:Jordan–Hölder theorem
3635:
3508:
3403:
3362:
3189:
3148:
3070:
2960:
2834:
2791:
2767:
2710:
2619:
2430:
2410:
2300:
2150:
2001:
1968:
1893:
1790:
1754:
1679:
1591:
1387:
1377:
1330:
1323:
1293:
1260:
1258:{\displaystyle F_{m}}
1232:
1193:
1153:
1126:
1068:
1000:
920:
913:
494:
469:
432:
7115:Properties of groups
6954:improve this article
6678:Supersolvable groups
6533:are solvable, their
6525:; in particular, if
6284:
6264:
6238:
6218:
6162:
6113:
6034:
5951:
5762:
5729:
5718:Borel subgroup in GL
5688:
5668:
5638:
5608:
5588:
5564:
5509:
5453:
5316:
5266:
5246:
5214:
5156:
5136:
5070:
5050:
5013:
4957:
4934:
4906:
4884:
4847:
4812:
4786:
4760:
4710:
4681:
4651:
4508:
4479:
4451:
4429:
4385:
4263:
4243:Barrington's theorem
4239:Abel–Ruffini theorem
4221:is not solvable for
4212:> 4, we see that
4070:
4013:
3985:
3965:
3931:
3901:
3818:
3781:More generally, all
3577:
3446:
3383:
3367:, defined using the
3198:
3159:
3080:
2978:
2843:
2800:
2780:
2719:
2628:
2441:
2419:
2309:
2159:
2012:
1977:
1902:
1801:
1766:
1688:
1610:
1392:
1335:
1310:
1291:{\displaystyle f(x)}
1273:
1242:
1202:
1162:
1135:
1077:
1015:
925:
887:
481:
456:
419:
6911:"p-solvable-groups"
6684:supersolvable group
4801:{\displaystyle a,b}
4775:{\displaystyle b=0}
4494:{\displaystyle B,U}
4468:{\displaystyle B/U}
4084:{\displaystyle G''}
3807:In particular, the
3645:commutator subgroup
1601:composition factors
1101:
866:polynomial equation
125:Group homomorphisms
35:Algebraic structure
6768:virtually solvable
6710:finitely generated
6619:Burnside's theorem
6613:Burnside's theorem
6553:are solvable, and
6535:semidirect product
6486:are solvable, the
6478:In particular, if
6323:semidirect product
6296:
6270:
6250:
6224:
6201:
6195:
6145:
6091:
6017:
5934:
5924:
5832:
5745:
5704:
5674:
5654:
5624:
5594:
5570:
5539:
5492:
5486:
5436:
5302:
5252:
5232:
5200:
5142:
5114:
5056:
5036:
4996:
4990:
4940:
4920:
4892:
4870:
4833:
4798:
4772:
4746:
4696:
4667:
4634:
4628:
4580:
4541:
4491:
4465:
4437:
4412:
4371:
4361:
4306:
4081:
4053:
3999:{\displaystyle G'}
3996:
3971:
3943:{\displaystyle -1}
3940:
3917:
3884:
3743:under addition is
3706:composition series
3693:= 1 is called the
3630:
3503:
3398:
3369:semidirect product
3357:
3184:
3143:
3065:
2955:
2829:
2786:
2762:
2705:
2624:with group action
2614:
2425:
2405:
2295:
2155:with group action
2145:
1996:
1963:
1898:with group action
1888:
1785:
1761:minimal polynomial
1749:
1684:with group action
1674:
1586:
1372:
1318:
1288:
1255:
1227:
1188:
1148:
1121:
1080:
1063:
995:
908:
875:the corresponding
844:terminates in the
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
7073:978-0-387-94285-8
7060:Rotman, Joseph J.
7030:
7029:
7022:
7004:
6835:Prosolvable group
6790:hypoabelian group
6772:virtually abelian
6607:(direct) products
6577:is also solvable.
6537:is also solvable.
6440:is solvable, and
6399:is a subgroup of
6395:is solvable, and
6273:{\displaystyle S}
6227:{\displaystyle T}
5848:
5677:{\displaystyle B}
5597:{\displaystyle G}
5573:{\displaystyle G}
5255:{\displaystyle b}
5145:{\displaystyle B}
5059:{\displaystyle B}
4943:{\displaystyle B}
4319:
4229:> 4 there are
4118:alternating group
3974:{\displaystyle G}
3897:where the kernel
3803:Quaternion groups
3302:
3282:
2939:
2918:
2891:
2789:{\displaystyle 1}
2755:
2745:
2688:
2556:
2546:
2483:
2473:
2428:{\displaystyle 1}
2278:
2257:
2119:
2109:
2054:
2044:
1946:
1939:
1921:
1863:
1841:
1831:
1732:
1725:
1707:
1640:
1541:
1531:
1478:
1468:
1447:
1437:
1416:
1370:
1362:
1352:
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
7122:
7088:
7076:
7055:
7025:
7018:
7014:
7011:
7005:
7003:
6969:"Solvable group"
6962:
6938:
6930:
6919:
6918:
6915:Group props wiki
6907:
6901:
6889:
6883:
6871:
6865:
6864:
6862:
6851:
6673:Related concepts
6573:with respect to
6373:
6355:
6341:are (start with
6321:are cyclic is a
6319:-Sylow subgroups
6305:
6303:
6302:
6297:
6279:
6277:
6276:
6271:
6259:
6257:
6256:
6251:
6233:
6231:
6230:
6225:
6210:
6208:
6207:
6202:
6200:
6199:
6154:
6152:
6151:
6146:
6144:
6143:
6128:
6127:
6100:
6098:
6097:
6092:
6087:
6086:
6081:
6072:
6071:
6066:
6057:
6056:
6051:
6026:
6024:
6023:
6018:
6016:
6015:
6010:
6001:
6000:
5995:
5986:
5985:
5980:
5971:
5970:
5961:
5943:
5941:
5940:
5935:
5933:
5929:
5928:
5859:
5858:
5849:
5846:
5841:
5837:
5836:
5754:
5752:
5751:
5746:
5744:
5743:
5713:
5711:
5710:
5705:
5703:
5702:
5683:
5681:
5680:
5675:
5663:
5661:
5660:
5655:
5653:
5652:
5633:
5631:
5630:
5625:
5623:
5622:
5603:
5601:
5600:
5595:
5579:
5577:
5576:
5571:
5548:
5546:
5545:
5540:
5501:
5499:
5498:
5493:
5491:
5490:
5445:
5443:
5442:
5437:
5435:
5409:
5350:
5311:
5309:
5308:
5303:
5261:
5259:
5258:
5253:
5241:
5239:
5238:
5233:
5209:
5207:
5206:
5201:
5196:
5195:
5190:
5181:
5180:
5175:
5163:
5151:
5149:
5148:
5143:
5123:
5121:
5120:
5115:
5113:
5112:
5107:
5098:
5097:
5092:
5080:
5065:
5063:
5062:
5057:
5045:
5043:
5042:
5037:
5032:
5005:
5003:
5002:
4997:
4995:
4994:
4949:
4947:
4946:
4941:
4929:
4927:
4926:
4921:
4919:
4901:
4899:
4898:
4893:
4891:
4879:
4877:
4876:
4871:
4866:
4842:
4840:
4839:
4834:
4807:
4805:
4804:
4799:
4781:
4779:
4778:
4773:
4755:
4753:
4752:
4747:
4739:
4738:
4733:
4724:
4723:
4718:
4705:
4703:
4702:
4697:
4676:
4674:
4673:
4668:
4666:
4665:
4643:
4641:
4640:
4635:
4633:
4632:
4585:
4584:
4546:
4545:
4500:
4498:
4497:
4492:
4474:
4472:
4471:
4466:
4461:
4446:
4444:
4443:
4438:
4436:
4421:
4419:
4418:
4413:
4408:
4400:
4399:
4380:
4378:
4377:
4372:
4370:
4366:
4365:
4320:
4317:
4315:
4311:
4310:
4090:
4088:
4087:
4082:
4080:
4062:
4060:
4059:
4054:
4046:
4035:
4005:
4003:
4002:
3997:
3995:
3980:
3978:
3977:
3972:
3959:Group extensions
3954:Group extensions
3949:
3947:
3946:
3941:
3926:
3924:
3923:
3918:
3913:
3908:
3893:
3891:
3890:
3885:
3874:
3869:
3858:
3853:
3836:
3831:
3809:quaternion group
3783:nilpotent groups
3777:Nilpotent groups
3639:
3637:
3636:
3631:
3620:
3619:
3601:
3600:
3512:
3510:
3509:
3504:
3496:
3495:
3477:
3476:
3464:
3463:
3426:subnormal series
3407:
3405:
3404:
3399:
3397:
3396:
3391:
3366:
3364:
3363:
3358:
3356:
3355:
3343:
3342:
3312:
3311:
3300:
3299:
3280:
3273:
3272:
3267:
3258:
3257:
3252:
3237:
3236:
3231:
3225:
3224:
3215:
3214:
3209:
3193:
3191:
3190:
3185:
3183:
3182:
3152:
3150:
3149:
3144:
3139:
3138:
3134:
3074:
3072:
3071:
3066:
3061:
3060:
3056:
3028:
3027:
3023:
2964:
2962:
2961:
2956:
2937:
2916:
2889:
2885:
2884:
2875:
2874:
2865:
2864:
2855:
2854:
2838:
2836:
2835:
2830:
2828:
2827:
2818:
2817:
2795:
2793:
2792:
2787:
2771:
2769:
2768:
2763:
2761:
2757:
2756:
2751:
2746:
2741:
2731:
2730:
2714:
2712:
2711:
2706:
2698:
2697:
2686:
2679:
2678:
2674:
2640:
2639:
2623:
2621:
2620:
2615:
2610:
2605:
2597:
2593:
2592:
2588:
2587:
2583:
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2523:
2516:
2515:
2511:
2484:
2479:
2474:
2469:
2464:
2454:
2434:
2432:
2431:
2426:
2414:
2412:
2411:
2406:
2389:
2375:
2374:
2347:
2346:
2334:
2333:
2321:
2320:
2304:
2302:
2301:
2296:
2288:
2287:
2276:
2255:
2251:
2250:
2246:
2206:
2202:
2201:
2197:
2171:
2170:
2154:
2152:
2151:
2146:
2141:
2136:
2128:
2124:
2120:
2115:
2110:
2105:
2100:
2095:
2090:
2086:
2085:
2081:
2055:
2050:
2045:
2040:
2035:
2025:
2005:
2003:
2002:
1997:
1989:
1988:
1972:
1970:
1969:
1964:
1956:
1955:
1944:
1940:
1935:
1927:
1923:
1922:
1917:
1897:
1895:
1894:
1889:
1884:
1879:
1868:
1864:
1859:
1850:
1846:
1842:
1837:
1832:
1827:
1814:
1794:
1792:
1791:
1786:
1778:
1777:
1758:
1756:
1755:
1750:
1742:
1741:
1730:
1726:
1721:
1713:
1709:
1708:
1703:
1683:
1681:
1680:
1675:
1670:
1665:
1654:
1649:
1645:
1641:
1636:
1623:
1595:
1593:
1592:
1587:
1585:
1581:
1574:
1573:
1569:
1542:
1537:
1532:
1527:
1522:
1514:
1510:
1509:
1505:
1479:
1474:
1469:
1464:
1459:
1448:
1443:
1438:
1433:
1428:
1417:
1412:
1407:
1399:
1381:
1379:
1378:
1373:
1371:
1369:
1364:
1363:
1358:
1353:
1348:
1345:
1327:
1325:
1324:
1319:
1317:
1297:
1295:
1294:
1289:
1264:
1262:
1261:
1256:
1254:
1253:
1236:
1234:
1233:
1228:
1226:
1225:
1197:
1195:
1194:
1189:
1181:
1180:
1179:
1178:
1157:
1155:
1154:
1149:
1147:
1146:
1130:
1128:
1127:
1122:
1120:
1119:
1100:
1099:
1098:
1088:
1072:
1070:
1069:
1064:
1059:
1058:
1046:
1045:
1027:
1026:
1004:
1002:
1001:
996:
988:
987:
969:
968:
956:
955:
943:
942:
917:
915:
914:
909:
846:trivial subgroup
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
65:
54:
41:
30:
21:
7130:
7129:
7125:
7124:
7123:
7121:
7120:
7119:
7110:Solvable groups
7100:
7099:
7083:
7074:
7058:
7047:(67): 347–366,
7033:
7026:
7015:
7009:
7006:
6963:
6961:
6951:
6939:
6928:
6923:
6922:
6909:
6908:
6904:
6891:Rotman (1995),
6890:
6886:
6873:Rotman (1995),
6872:
6868:
6860:
6853:
6852:
6848:
6843:
6831:
6819:
6780:
6760:
6718:
6700:) if it has an
6686:
6680:
6675:
6621:
6615:
6561:-set, then the
6503:group extension
6385:
6369:
6351:
6335:
6312:
6282:
6281:
6262:
6261:
6236:
6235:
6216:
6215:
6194:
6193:
6188:
6182:
6181:
6176:
6166:
6160:
6159:
6135:
6119:
6111:
6110:
6107:
6076:
6061:
6046:
6032:
6031:
6005:
5990:
5975:
5962:
5949:
5948:
5923:
5922:
5917:
5912:
5906:
5905:
5900:
5895:
5889:
5888:
5883:
5878:
5868:
5863:
5850:
5831:
5830:
5825:
5820:
5814:
5813:
5808:
5803:
5797:
5796:
5791:
5786:
5776:
5771:
5760:
5759:
5735:
5727:
5726:
5723:
5721:
5694:
5686:
5685:
5666:
5665:
5644:
5636:
5635:
5614:
5606:
5605:
5586:
5585:
5562:
5561:
5555:
5553:Borel subgroups
5507:
5506:
5485:
5484:
5479:
5473:
5472:
5467:
5457:
5451:
5450:
5428:
5402:
5343:
5314:
5313:
5312:. This implies
5264:
5263:
5244:
5243:
5212:
5211:
5185:
5170:
5154:
5153:
5134:
5133:
5130:
5102:
5087:
5068:
5067:
5048:
5047:
5011:
5010:
4989:
4988:
4983:
4977:
4976:
4971:
4961:
4955:
4954:
4932:
4931:
4904:
4903:
4882:
4881:
4845:
4844:
4810:
4809:
4784:
4783:
4758:
4757:
4728:
4713:
4708:
4707:
4679:
4678:
4657:
4649:
4648:
4627:
4626:
4621:
4615:
4614:
4600:
4590:
4579:
4578:
4573:
4567:
4566:
4561:
4551:
4540:
4539:
4534:
4528:
4527:
4522:
4512:
4506:
4505:
4477:
4476:
4449:
4448:
4427:
4426:
4425:for some field
4391:
4383:
4382:
4360:
4359:
4354:
4348:
4347:
4342:
4332:
4327:
4305:
4304:
4299:
4293:
4292:
4287:
4277:
4272:
4261:
4260:
4254:
4252:
4249:Subgroups of GL
4220:
4207:
4198:
4189:
4182:
4175:
4164:
4157:
4150:
4142:
4130:
4115:
4108:
4101:symmetric group
4097:
4073:
4068:
4067:
4039:
4028:
4011:
4010:
3988:
3983:
3982:
3963:
3962:
3956:
3929:
3928:
3899:
3898:
3816:
3815:
3805:
3799:are nilpotent.
3779:
3770:
3765:
3666:, the quotient
3657:normal subgroup
3605:
3586:
3575:
3574:
3554:
3542:
3535:
3525:
3487:
3468:
3455:
3444:
3443:
3414:
3386:
3381:
3380:
3347:
3331:
3303:
3262:
3247:
3226:
3216:
3204:
3196:
3195:
3174:
3157:
3156:
3116:
3078:
3077:
3038:
3005:
2976:
2975:
2876:
2866:
2856:
2846:
2841:
2840:
2819:
2809:
2798:
2797:
2778:
2777:
2739:
2735:
2722:
2717:
2716:
2689:
2653:
2631:
2626:
2625:
2565:
2561:
2493:
2492:
2488:
2459:
2455:
2439:
2438:
2417:
2416:
2366:
2338:
2325:
2312:
2307:
2306:
2279:
2210:
2176:
2172:
2162:
2157:
2156:
2063:
2059:
2030:
2026:
2010:
2009:
1980:
1975:
1974:
1947:
1912:
1908:
1900:
1899:
1815:
1799:
1798:
1769:
1764:
1763:
1733:
1698:
1694:
1686:
1685:
1624:
1608:
1607:
1551:
1550:
1546:
1487:
1483:
1390:
1389:
1346:
1333:
1332:
1308:
1307:
1304:
1271:
1270:
1267:splitting field
1245:
1240:
1239:
1211:
1200:
1199:
1170:
1165:
1160:
1159:
1138:
1133:
1132:
1105:
1090:
1075:
1074:
1050:
1031:
1018:
1013:
1012:
979:
960:
947:
934:
923:
922:
885:
884:
868:is solvable in
854:
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
479:
478:
454:
453:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
7128:
7126:
7118:
7117:
7112:
7102:
7101:
7098:
7097:
7092:
7082:
7081:External links
7079:
7078:
7077:
7072:
7056:
7043:, New Series,
7028:
7027:
6942:
6940:
6933:
6927:
6924:
6921:
6920:
6902:
6884:
6866:
6845:
6844:
6842:
6839:
6838:
6837:
6830:
6827:
6818:
6815:
6779:
6776:
6759:
6756:
6755:
6754:
6716:
6682:Main article:
6679:
6676:
6674:
6671:
6617:Main article:
6614:
6611:
6587:derived length
6579:
6578:
6563:wreath product
6539:
6538:
6499:
6498:
6488:direct product
6476:
6457:
6408:
6384:
6381:
6380:
6379:
6362:
6361:
6334:
6331:
6311:
6308:
6295:
6292:
6289:
6269:
6249:
6246:
6243:
6223:
6198:
6192:
6189:
6187:
6184:
6183:
6180:
6177:
6175:
6172:
6171:
6169:
6142:
6138:
6134:
6131:
6126:
6122:
6118:
6106:
6103:
6090:
6085:
6080:
6075:
6070:
6065:
6060:
6055:
6050:
6045:
6042:
6039:
6014:
6009:
6004:
5999:
5994:
5989:
5984:
5979:
5974:
5969:
5965:
5960:
5956:
5932:
5927:
5921:
5918:
5916:
5913:
5911:
5908:
5907:
5904:
5901:
5899:
5896:
5894:
5891:
5890:
5887:
5884:
5882:
5879:
5877:
5874:
5873:
5871:
5866:
5862:
5857:
5853:
5844:
5840:
5835:
5829:
5826:
5824:
5821:
5819:
5816:
5815:
5812:
5809:
5807:
5804:
5802:
5799:
5798:
5795:
5792:
5790:
5787:
5785:
5782:
5781:
5779:
5774:
5770:
5767:
5742:
5738:
5734:
5722:
5719:
5716:
5701:
5697:
5693:
5673:
5651:
5647:
5643:
5621:
5617:
5613:
5593:
5582:Borel subgroup
5569:
5554:
5551:
5549:in the group.
5538:
5535:
5532:
5529:
5526:
5523:
5520:
5517:
5514:
5489:
5483:
5480:
5478:
5475:
5474:
5471:
5468:
5466:
5463:
5462:
5460:
5434:
5431:
5427:
5424:
5421:
5418:
5415:
5412:
5408:
5405:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5359:
5356:
5353:
5349:
5346:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5321:
5301:
5298:
5295:
5292:
5289:
5286:
5283:
5280:
5277:
5274:
5271:
5251:
5231:
5228:
5225:
5222:
5219:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
5166:
5162:
5141:
5129:
5126:
5111:
5106:
5101:
5096:
5091:
5086:
5083:
5079:
5075:
5055:
5035:
5031:
5027:
5024:
5021:
5018:
4993:
4987:
4984:
4982:
4979:
4978:
4975:
4972:
4970:
4967:
4966:
4964:
4939:
4918:
4914:
4911:
4890:
4869:
4865:
4861:
4858:
4855:
4852:
4832:
4829:
4826:
4823:
4820:
4817:
4797:
4794:
4791:
4771:
4768:
4765:
4745:
4742:
4737:
4732:
4727:
4722:
4717:
4695:
4692:
4689:
4686:
4664:
4660:
4656:
4631:
4625:
4622:
4620:
4617:
4616:
4613:
4610:
4607:
4604:
4601:
4599:
4596:
4595:
4593:
4588:
4583:
4577:
4574:
4572:
4569:
4568:
4565:
4562:
4560:
4557:
4556:
4554:
4549:
4544:
4538:
4535:
4533:
4530:
4529:
4526:
4523:
4521:
4518:
4517:
4515:
4490:
4487:
4484:
4464:
4460:
4456:
4435:
4411:
4407:
4403:
4398:
4394:
4390:
4369:
4364:
4358:
4355:
4353:
4350:
4349:
4346:
4343:
4341:
4338:
4337:
4335:
4330:
4326:
4323:
4314:
4309:
4303:
4300:
4298:
4295:
4294:
4291:
4288:
4286:
4283:
4282:
4280:
4275:
4271:
4268:
4253:
4250:
4247:
4216:
4203:
4194:
4187:
4180:
4173:
4162:
4155:
4148:
4141:
4138:
4129:
4126:
4113:
4106:
4096:
4093:
4079:
4076:
4052:
4049:
4045:
4042:
4038:
4034:
4031:
4027:
4024:
4021:
4018:
3994:
3991:
3970:
3955:
3952:
3939:
3936:
3916:
3912:
3907:
3883:
3880:
3877:
3873:
3868:
3864:
3861:
3857:
3852:
3848:
3845:
3842:
3839:
3835:
3830:
3826:
3823:
3804:
3801:
3778:
3775:
3769:
3768:Abelian groups
3766:
3764:
3761:
3695:derived length
3676:if and only if
3641:
3640:
3629:
3626:
3623:
3618:
3615:
3612:
3608:
3604:
3599:
3596:
3593:
3589:
3585:
3582:
3568:derived series
3549:
3540:
3533:
3520:
3514:
3513:
3502:
3499:
3494:
3490:
3486:
3483:
3480:
3475:
3471:
3467:
3462:
3458:
3454:
3451:
3413:
3410:
3395:
3390:
3373:direct product
3354:
3350:
3346:
3341:
3338:
3334:
3330:
3327:
3324:
3321:
3318:
3315:
3310:
3306:
3298:
3295:
3292:
3289:
3286:
3279:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3243:
3240:
3235:
3230:
3223:
3219:
3213:
3208:
3203:
3181:
3177:
3173:
3170:
3167:
3164:
3142:
3137:
3133:
3129:
3126:
3123:
3119:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3064:
3059:
3055:
3051:
3048:
3045:
3041:
3037:
3034:
3031:
3026:
3022:
3018:
3015:
3012:
3008:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2954:
2951:
2948:
2945:
2942:
2936:
2933:
2930:
2927:
2924:
2921:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2888:
2883:
2879:
2873:
2869:
2863:
2859:
2853:
2849:
2826:
2822:
2816:
2812:
2808:
2805:
2785:
2774:
2773:
2760:
2754:
2749:
2744:
2738:
2734:
2729:
2725:
2704:
2701:
2696:
2692:
2685:
2682:
2677:
2673:
2669:
2666:
2663:
2660:
2656:
2652:
2649:
2646:
2643:
2638:
2634:
2613:
2609:
2604:
2600:
2596:
2591:
2586:
2582:
2578:
2575:
2572:
2568:
2564:
2560:
2555:
2550:
2545:
2540:
2536:
2531:
2526:
2522:
2519:
2514:
2510:
2506:
2503:
2500:
2496:
2491:
2487:
2482:
2477:
2472:
2467:
2463:
2458:
2453:
2450:
2447:
2436:
2424:
2404:
2401:
2398:
2395:
2392:
2388:
2384:
2381:
2378:
2373:
2369:
2365:
2362:
2359:
2356:
2353:
2350:
2345:
2341:
2337:
2332:
2328:
2324:
2319:
2315:
2294:
2291:
2286:
2282:
2275:
2272:
2269:
2266:
2263:
2260:
2254:
2249:
2245:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2213:
2209:
2205:
2200:
2196:
2192:
2189:
2186:
2183:
2179:
2175:
2169:
2165:
2144:
2140:
2135:
2131:
2127:
2123:
2118:
2113:
2108:
2103:
2099:
2094:
2089:
2084:
2080:
2076:
2073:
2070:
2066:
2062:
2058:
2053:
2048:
2043:
2038:
2034:
2029:
2024:
2021:
2018:
2007:
1995:
1992:
1987:
1983:
1962:
1959:
1954:
1950:
1943:
1938:
1933:
1930:
1926:
1920:
1915:
1911:
1907:
1887:
1883:
1878:
1874:
1871:
1867:
1862:
1857:
1854:
1849:
1845:
1840:
1835:
1830:
1825:
1822:
1818:
1813:
1810:
1807:
1796:
1784:
1781:
1776:
1772:
1748:
1745:
1740:
1736:
1729:
1724:
1719:
1716:
1712:
1706:
1701:
1697:
1693:
1673:
1669:
1664:
1660:
1657:
1653:
1648:
1644:
1639:
1634:
1631:
1627:
1622:
1619:
1616:
1584:
1580:
1577:
1572:
1568:
1564:
1561:
1558:
1554:
1549:
1545:
1540:
1535:
1530:
1525:
1521:
1517:
1513:
1508:
1504:
1500:
1497:
1494:
1490:
1486:
1482:
1477:
1472:
1467:
1462:
1458:
1454:
1451:
1446:
1441:
1436:
1431:
1427:
1423:
1420:
1415:
1410:
1406:
1402:
1398:
1368:
1361:
1356:
1351:
1343:
1340:
1316:
1303:
1300:
1299:
1298:
1287:
1284:
1281:
1278:
1252:
1248:
1237:
1224:
1221:
1218:
1214:
1210:
1207:
1187:
1184:
1177:
1173:
1168:
1145:
1141:
1118:
1115:
1112:
1108:
1104:
1097:
1093:
1087:
1083:
1062:
1057:
1053:
1049:
1044:
1041:
1038:
1034:
1030:
1025:
1021:
994:
991:
986:
982:
978:
975:
972:
967:
963:
959:
954:
950:
946:
941:
937:
933:
930:
907:
904:
901:
898:
895:
892:
881:characteristic
873:if and only if
853:
850:
842:derived series
834:abelian groups
822:solvable group
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
24:
18:Derived length
14:
13:
10:
9:
6:
4:
3:
2:
7127:
7116:
7113:
7111:
7108:
7107:
7105:
7096:
7093:
7091:
7085:
7084:
7080:
7075:
7069:
7065:
7061:
7057:
7054:
7050:
7046:
7042:
7041:
7036:
7035:Malcev, A. I.
7032:
7031:
7024:
7021:
7013:
7002:
6999:
6995:
6992:
6988:
6985:
6981:
6978:
6974:
6971: –
6970:
6966:
6965:Find sources:
6959:
6955:
6949:
6948:
6943:This article
6941:
6937:
6932:
6931:
6925:
6916:
6912:
6906:
6903:
6900:
6897:, p. 102, at
6896:
6895:
6888:
6885:
6882:
6879:, p. 102, at
6878:
6877:
6870:
6867:
6863:. p. 45.
6859:
6858:
6850:
6847:
6840:
6836:
6833:
6832:
6828:
6826:
6824:
6816:
6814:
6812:
6808:
6804:
6800:
6796:
6792:
6791:
6785:
6777:
6775:
6773:
6769:
6765:
6757:
6752:
6748:
6744:
6740:
6739:supersolvable
6736:
6732:
6728:
6725:
6724:
6723:
6720:
6715:
6711:
6707:
6703:
6699:
6695:
6694:supersolvable
6691:
6685:
6677:
6672:
6670:
6669:is solvable.
6668:
6664:
6661:
6657:
6653:
6649:
6648:prime numbers
6645:
6641:
6637:
6634:
6630:
6626:
6620:
6610:
6608:
6604:
6600:
6596:
6592:
6588:
6584:
6576:
6572:
6568:
6564:
6560:
6556:
6552:
6548:
6544:
6543:
6542:
6536:
6532:
6528:
6524:
6520:
6516:
6512:
6508:
6507:
6506:
6504:
6496:
6492:
6489:
6485:
6481:
6477:
6475:are solvable.
6474:
6470:
6466:
6462:
6458:
6455:
6451:
6447:
6443:
6439:
6435:
6431:
6427:
6424:
6421:
6417:
6413:
6409:
6406:
6402:
6398:
6394:
6390:
6389:
6388:
6382:
6377:
6372:
6367:
6366:
6365:
6359:
6354:
6348:
6347:
6346:
6344:
6340:
6332:
6330:
6328:
6324:
6320:
6318:
6309:
6307:
6293:
6290:
6287:
6267:
6247:
6244:
6241:
6221:
6211:
6196:
6190:
6185:
6178:
6173:
6167:
6156:
6140:
6136:
6132:
6129:
6124:
6120:
6116:
6104:
6101:
6083:
6073:
6068:
6058:
6053:
6040:
6037:
6028:
6012:
6002:
5997:
5987:
5982:
5972:
5967:
5963:
5958:
5954:
5944:
5930:
5925:
5919:
5914:
5909:
5902:
5897:
5892:
5885:
5880:
5875:
5869:
5864:
5860:
5855:
5851:
5842:
5838:
5833:
5827:
5822:
5817:
5810:
5805:
5800:
5793:
5788:
5783:
5777:
5772:
5768:
5765:
5756:
5740:
5736:
5732:
5717:
5715:
5699:
5695:
5691:
5671:
5649:
5645:
5641:
5619:
5615:
5611:
5591:
5583:
5567:
5560:
5552:
5550:
5533:
5530:
5527:
5521:
5515:
5502:
5487:
5481:
5476:
5469:
5464:
5458:
5447:
5432:
5429:
5425:
5422:
5419:
5416:
5413:
5406:
5403:
5393:
5390:
5387:
5381:
5375:
5366:
5363:
5360:
5354:
5347:
5344:
5340:
5337:
5328:
5325:
5322:
5299:
5296:
5293:
5287:
5278:
5275:
5272:
5249:
5226:
5223:
5220:
5192:
5182:
5177:
5164:
5139:
5127:
5125:
5109:
5099:
5094:
5084:
5081:
5077:
5073:
5053:
5033:
5029:
5025:
5022:
5019:
5016:
5006:
4991:
4985:
4980:
4973:
4968:
4962:
4951:
4937:
4912:
4909:
4867:
4863:
4859:
4856:
4853:
4850:
4830:
4827:
4824:
4821:
4818:
4815:
4795:
4792:
4789:
4782:). For fixed
4769:
4766:
4763:
4743:
4740:
4735:
4725:
4720:
4693:
4690:
4687:
4684:
4662:
4658:
4654:
4644:
4629:
4623:
4618:
4611:
4608:
4605:
4602:
4597:
4591:
4586:
4581:
4575:
4570:
4563:
4558:
4552:
4547:
4542:
4536:
4531:
4524:
4519:
4513:
4502:
4488:
4485:
4482:
4462:
4458:
4454:
4422:
4396:
4392:
4388:
4367:
4362:
4356:
4351:
4344:
4339:
4333:
4328:
4324:
4321:
4312:
4307:
4301:
4296:
4289:
4284:
4278:
4273:
4269:
4266:
4257:
4248:
4246:
4244:
4240:
4236:
4232:
4228:
4224:
4219:
4215:
4211:
4206:
4202:
4197:
4193:
4186:
4179:
4172:
4168:
4161:
4154:
4147:
4139:
4137:
4135:
4127:
4125:
4123:
4119:
4112:
4105:
4102:
4094:
4092:
4077:
4074:
4063:
4050:
4043:
4040:
4032:
4029:
4022:
4016:
4007:
3992:
3989:
3968:
3960:
3953:
3951:
3937:
3934:
3914:
3910:
3894:
3881:
3875:
3871:
3862:
3859:
3855:
3843:
3837:
3833:
3821:
3812:
3810:
3802:
3800:
3798:
3796:
3791:
3789:
3784:
3776:
3774:
3767:
3762:
3760:
3758:
3754:
3750:
3746:
3742:
3738:
3734:
3730:
3726:
3722:
3718:
3715:
3711:
3710:cyclic groups
3707:
3702:
3700:
3696:
3692:
3688:
3684:
3680:
3677:
3673:
3669:
3665:
3661:
3658:
3654:
3650:
3646:
3627:
3624:
3621:
3613:
3606:
3602:
3594:
3587:
3583:
3580:
3573:
3572:
3571:
3569:
3564:
3562:
3559:= 1, 2, ...,
3558:
3552:
3548:
3544:
3536:
3529:
3523:
3519:
3516:meaning that
3500:
3497:
3492:
3488:
3484:
3481:
3478:
3473:
3469:
3465:
3460:
3456:
3452:
3449:
3442:
3441:
3440:
3439:
3435:
3431:
3430:factor groups
3427:
3423:
3419:
3411:
3409:
3393:
3378:
3377:cyclic groups
3374:
3370:
3352:
3348:
3344:
3339:
3336:
3332:
3328:
3325:
3322:
3316:
3308:
3304:
3277:
3269:
3259:
3254:
3241:
3233:
3221:
3217:
3211:
3179:
3175:
3171:
3168:
3165:
3162:
3153:
3140:
3135:
3131:
3127:
3124:
3121:
3117:
3113:
3107:
3101:
3098:
3092:
3086:
3083:
3075:
3062:
3057:
3053:
3049:
3046:
3043:
3039:
3035:
3029:
3024:
3020:
3016:
3013:
3010:
3006:
2999:
2996:
2990:
2984:
2981:
2973:
2971:
2966:
2952:
2949:
2946:
2943:
2940:
2934:
2931:
2928:
2925:
2922:
2919:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2886:
2881:
2877:
2871:
2867:
2861:
2857:
2851:
2847:
2824:
2820:
2814:
2810:
2806:
2803:
2783:
2758:
2752:
2747:
2742:
2736:
2732:
2727:
2723:
2702:
2699:
2694:
2690:
2683:
2680:
2675:
2671:
2667:
2664:
2661:
2658:
2654:
2650:
2644:
2636:
2632:
2611:
2607:
2598:
2594:
2589:
2584:
2580:
2576:
2573:
2570:
2566:
2562:
2553:
2548:
2543:
2529:
2524:
2520:
2517:
2512:
2508:
2504:
2501:
2498:
2494:
2489:
2480:
2475:
2470:
2456:
2437:
2422:
2399:
2396:
2393:
2386:
2379:
2376:
2371:
2367:
2360:
2357:
2354:
2351:
2348:
2343:
2339:
2335:
2330:
2326:
2322:
2317:
2313:
2292:
2289:
2284:
2280:
2273:
2270:
2267:
2264:
2261:
2258:
2252:
2247:
2243:
2239:
2236:
2233:
2227:
2224:
2221:
2215:
2211:
2207:
2203:
2198:
2194:
2190:
2187:
2184:
2181:
2177:
2173:
2167:
2163:
2142:
2138:
2129:
2125:
2116:
2111:
2106:
2092:
2087:
2082:
2078:
2074:
2071:
2068:
2064:
2060:
2051:
2046:
2041:
2027:
2008:
1993:
1990:
1985:
1981:
1960:
1957:
1952:
1948:
1941:
1936:
1931:
1928:
1924:
1918:
1913:
1909:
1905:
1885:
1881:
1872:
1816:
1797:
1782:
1779:
1774:
1770:
1762:
1746:
1743:
1738:
1734:
1727:
1722:
1717:
1714:
1710:
1704:
1699:
1695:
1691:
1671:
1667:
1658:
1625:
1606:
1605:
1604:
1602:
1596:
1582:
1578:
1575:
1570:
1566:
1562:
1559:
1556:
1552:
1547:
1538:
1533:
1528:
1515:
1511:
1506:
1502:
1498:
1495:
1492:
1488:
1484:
1475:
1470:
1465:
1452:
1444:
1439:
1434:
1421:
1413:
1400:
1386:
1382:
1366:
1359:
1354:
1349:
1341:
1338:
1329:
1301:
1282:
1276:
1268:
1250:
1246:
1238:
1222:
1219:
1216:
1212:
1208:
1205:
1185:
1182:
1175:
1171:
1166:
1143:
1139:
1116:
1113:
1110:
1106:
1102:
1095:
1091:
1085:
1081:
1055:
1051:
1042:
1039:
1036:
1032:
1028:
1023:
1019:
1011:
1010:
1009:
1005:
992:
989:
984:
980:
976:
973:
970:
965:
961:
957:
952:
948:
944:
939:
935:
931:
928:
919:
902:
896:
893:
890:
882:
878:
874:
871:
867:
863:
859:
858:Galois theory
851:
849:
847:
843:
839:
835:
831:
827:
826:soluble group
823:
819:
815:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
31:
19:
7063:
7044:
7040:Mat. Sbornik
7038:
7016:
7010:January 2008
7007:
6997:
6990:
6983:
6976:
6964:
6952:Please help
6947:verification
6944:
6914:
6905:
6899:Google Books
6894:Theorem 5.16
6892:
6887:
6881:Google Books
6876:Theorem 5.15
6874:
6869:
6857:Field Theory
6856:
6849:
6820:
6806:
6802:
6798:
6794:
6788:
6783:
6781:
6767:
6763:
6761:
6746:
6721:
6713:
6701:
6698:supersoluble
6697:
6693:
6689:
6687:
6666:
6660:non-negative
6655:
6651:
6643:
6639:
6635:
6629:finite group
6624:
6622:
6590:
6582:
6580:
6574:
6570:
6566:
6558:
6554:
6550:
6546:
6540:
6530:
6526:
6522:
6518:
6514:
6510:
6500:
6497:is solvable.
6494:
6490:
6483:
6479:
6472:
6468:
6464:
6460:
6456:is solvable.
6453:
6449:
6445:
6441:
6437:
6429:
6425:
6419:
6416:homomorphism
6411:
6407:is solvable.
6404:
6400:
6396:
6392:
6386:
6363:
6342:
6338:
6336:
6316:
6313:
6213:
6158:
6108:
6030:
5946:
5758:
5724:
5556:
5504:
5449:
5131:
5008:
4953:
4646:
4504:
4424:
4259:
4255:
4234:
4226:
4222:
4217:
4213:
4209:
4204:
4200:
4195:
4191:
4184:
4177:
4170:
4159:
4152:
4145:
4143:
4131:
4121:
4110:
4103:
4098:
4065:
4009:
3957:
3896:
3814:
3806:
3794:
3787:
3780:
3771:
3756:
3752:
3748:
3736:
3728:
3703:
3698:
3694:
3690:
3686:
3685:. The least
3682:
3678:
3671:
3667:
3663:
3659:
3652:
3648:
3642:
3565:
3560:
3556:
3550:
3546:
3538:
3537:, such that
3531:
3521:
3517:
3515:
3424:if it has a
3421:
3417:
3415:
3154:
3076:
2974:
2967:
2775:
1598:
1388:
1384:
1331:
1305:
1007:
921:
877:Galois group
855:
825:
821:
818:group theory
811:
640:
628:
616:
604:
592:
580:
568:
556:
327:
284:
271:
260:
249:
245:Cyclic group
200:
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
6811:Malcev 1949
6778:Hypoabelian
6706:uncountable
6603:subalgebras
6599:homomorphic
6333:OEIS values
4231:polynomials
4165:} (and the
4140:Non-example
3674:is abelian
1265:contains a
814:mathematics
530:Topological
369:alternating
7104:Categories
6980:newspapers
6926:References
6817:p-solvable
6797:such that
6766:is called
6743:polycyclic
6692:is called
6595:subvariety
6383:Properties
4233:of degree
4144:The group
3745:isomorphic
3689:such that
3655:and every
3420:is called
3412:Definition
1008:such that
852:Motivation
838:extensions
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
6735:nilpotent
6702:invariant
6291:×
6245:×
6130:×
6084:×
6074:×
6069:×
6059:×
6054:×
6041:⋊
6013:×
6003:×
5998:×
5988:×
5983:×
5973:≅
5903:∗
5886:∗
5881:∗
5828:∗
5811:∗
5806:∗
5794:∗
5789:∗
5784:∗
5522:×
5193:×
5183:×
5178:×
5165:⋊
5110:×
5100:×
5095:×
5085:≅
5023:−
4913:∈
4857:−
4741:⊂
4736:×
4726:×
4721:×
4691:≠
4548:⋅
4345:∗
4302:∗
4290:∗
4285:∗
4048:→
4037:→
4026:→
4020:→
3935:−
3879:→
3863:×
3847:→
3841:→
3825:→
3625:⋯
3622:▹
3603:▹
3584:▹
3485:◃
3482:⋯
3479:◃
3466:◃
3438:subgroups
3337:−
3305:φ
3260:×
3242:×
3222:φ
3218:⋊
3128:π
3050:π
3017:π
2950:≤
2944:≤
2929:≤
2923:≤
2908:≤
2896:≤
2733:−
2668:π
2599:≅
2577:π
2505:π
2397:−
2377:−
2268:≤
2262:≤
2240:π
2191:π
2130:≅
2075:π
1991:−
1932:∓
1914:±
1873:≅
1780:−
1718:∓
1700:±
1659:≅
1563:π
1516:⊆
1499:π
1453:⊆
1422:⊆
1401:⊆
1220:−
1209:∈
1183:−
1140:α
1114:−
1103:∈
1082:α
1052:α
1040:−
977:⊆
974:⋯
971:⊆
958:⊆
945:⊆
894:∈
701:Conformal
589:Euclidean
196:nilpotent
7062:(1995),
6829:See also
6762:A group
6747:solvable
6663:integers
6601:images,
6589:at most
6327:Z-groups
6310:Z-groups
5433:′
5407:′
5348:′
5242:acts on
4843:implies
4706:, hence
4677:implies
4078:″
4044:′
4033:″
3993:′
3763:Examples
3741:integers
3422:solvable
3416:A group
2776:, where
870:radicals
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
7053:0032644
6994:scholar
6854:Milne.
6823:p-group
6731:abelian
6665:, then
6593:form a
6493:×
6448:, then
6428:, then
6403:, then
6374:in the
6371:A056866
6356:in the
6353:A201733
5947:Notice
4318:,
4116:, (the
3797:-groups
3790:-groups
3543:
3434:abelian
3375:of the
2970:abelian
1302:Example
862:quintic
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
7070:
7051:
6996:
6989:
6982:
6975:
6967:
6784:finite
6727:cyclic
6650:, and
6638:where
6605:, and
6436:), if
6234:is an
6214:where
5847:
5557:For a
5210:where
5128:Remark
4902:since
4183:; and
3721:simple
3528:normal
3428:whose
3301:
3281:
2938:
2917:
2890:
2687:
2277:
2256:
1945:
1759:, and
1731:
1198:where
1073:where
836:using
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
7001:JSTOR
6987:books
6861:(PDF)
6841:Notes
6749:<
6745:<
6741:<
6737:<
6733:<
6729:<
6633:order
6627:is a
6557:is a
6418:from
6345:= 0)
6280:is a
5009:with
4122:every
3733:field
3717:order
3714:prime
1131:, so
830:group
828:is a
730:Sp(∞)
727:SU(∞)
140:image
7087:OEIS
7068:ISBN
6973:news
6696:(or
6658:are
6654:and
6646:are
6642:and
6569:and
6549:and
6529:and
6513:and
6482:and
6467:and
6423:onto
6376:OEIS
6358:OEIS
5634:and
5580:, a
4208:for
4176:and
4132:The
3981:and
3371:and
1269:for
820:, a
724:O(∞)
713:Loop
532:and
6956:by
6813:).
6631:of
6565:of
6545:If
6509:If
6410:If
6391:If
5725:In
5684:in
5262:by
5152:as
4381:of
3747:to
3739:of
3712:of
3662:of
3530:in
3526:is
824:or
812:In
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
7106::
7049:MR
7045:25
6913:.
6801:=
6636:pq
6505::
6329:.
5124:.
4245:.
4158:,
3950:.
3701:.
3563:.
3553:−1
3524:−1
1603::
848:.
615:U(
591:E(
579:O(
37:→
7023:)
7017:(
7012:)
7008:(
6998:·
6991:·
6984:·
6977:·
6950:.
6917:.
6807:G
6803:G
6799:G
6795:α
6764:G
6753:.
6717:4
6714:A
6690:G
6667:G
6656:b
6652:a
6644:q
6640:p
6625:G
6591:N
6583:N
6575:X
6571:H
6567:G
6559:G
6555:X
6551:H
6547:G
6531:H
6527:N
6523:G
6519:H
6517:/
6515:G
6511:H
6495:H
6491:G
6484:H
6480:G
6473:N
6471:/
6469:G
6465:N
6461:G
6454:N
6452:/
6450:G
6446:G
6442:N
6438:G
6430:H
6426:H
6420:G
6412:G
6405:H
6401:G
6397:H
6393:G
6378:)
6360:)
6343:n
6339:n
6317:p
6294:m
6288:m
6268:S
6248:n
6242:n
6222:T
6197:]
6191:S
6186:0
6179:0
6174:T
6168:[
6141:m
6137:L
6133:G
6125:n
6121:L
6117:G
6089:)
6079:F
6064:F
6049:F
6044:(
6038:U
6008:F
5993:F
5978:F
5968:1
5964:U
5959:/
5955:B
5931:}
5926:]
5920:1
5915:0
5910:0
5898:1
5893:0
5876:1
5870:[
5865:{
5861:=
5856:1
5852:U
5843:,
5839:}
5834:]
5823:0
5818:0
5801:0
5778:[
5773:{
5769:=
5766:B
5741:3
5737:L
5733:G
5720:3
5700:2
5696:L
5692:G
5672:B
5650:n
5646:L
5642:S
5620:n
5616:L
5612:G
5592:G
5568:G
5537:)
5534:c
5531:,
5528:a
5525:(
5519:)
5516:b
5513:(
5488:]
5482:c
5477:0
5470:b
5465:a
5459:[
5430:b
5426:a
5423:+
5420:b
5417:a
5414:=
5411:)
5404:b
5400:(
5397:)
5394:c
5391:,
5388:a
5385:(
5382:+
5379:)
5376:b
5373:(
5370:)
5367:c
5364:,
5361:a
5358:(
5355:=
5352:)
5345:b
5341:+
5338:b
5335:(
5332:)
5329:c
5326:,
5323:a
5320:(
5300:b
5297:a
5294:=
5291:)
5288:b
5285:(
5282:)
5279:c
5276:,
5273:a
5270:(
5250:b
5230:)
5227:c
5224:,
5221:a
5218:(
5198:)
5188:F
5173:F
5168:(
5161:F
5140:B
5105:F
5090:F
5082:U
5078:/
5074:B
5054:B
5034:a
5030:/
5026:b
5020:=
5017:d
4992:]
4986:1
4981:0
4974:d
4969:1
4963:[
4938:B
4917:F
4910:b
4889:F
4868:a
4864:/
4860:b
4854:=
4851:d
4831:0
4828:=
4825:b
4822:+
4819:d
4816:a
4796:b
4793:,
4790:a
4770:0
4767:=
4764:b
4744:B
4731:F
4716:F
4694:0
4688:c
4685:a
4663:2
4659:L
4655:G
4630:]
4624:c
4619:0
4612:b
4609:+
4606:d
4603:a
4598:a
4592:[
4587:=
4582:]
4576:1
4571:0
4564:d
4559:1
4553:[
4543:]
4537:c
4532:0
4525:b
4520:a
4514:[
4489:U
4486:,
4483:B
4463:U
4459:/
4455:B
4434:F
4410:)
4406:F
4402:(
4397:2
4393:L
4389:G
4368:}
4363:]
4357:1
4352:0
4340:1
4334:[
4329:{
4325:=
4322:U
4313:}
4308:]
4297:0
4279:[
4274:{
4270:=
4267:B
4251:2
4235:n
4227:n
4223:n
4218:n
4214:S
4210:n
4205:n
4201:S
4196:n
4192:A
4188:5
4185:A
4181:2
4178:C
4174:5
4171:A
4163:5
4160:S
4156:5
4153:A
4149:5
4146:S
4114:5
4111:A
4107:3
4104:S
4075:G
4051:1
4041:G
4030:G
4023:G
4017:1
3990:G
3969:G
3938:1
3915:2
3911:/
3906:Z
3882:1
3876:2
3872:/
3867:Z
3860:2
3856:/
3851:Z
3844:Q
3838:2
3834:/
3829:Z
3822:1
3795:p
3788:p
3757:Z
3753:Z
3749:Z
3737:Z
3729:n
3699:G
3691:G
3687:n
3683:H
3679:N
3672:N
3670:/
3668:H
3664:H
3660:N
3653:H
3649:G
3628:,
3617:)
3614:2
3611:(
3607:G
3598:)
3595:1
3592:(
3588:G
3581:G
3561:k
3557:j
3551:j
3547:G
3545:/
3541:j
3539:G
3534:j
3532:G
3522:j
3518:G
3501:G
3498:=
3493:k
3489:G
3474:1
3470:G
3461:0
3457:G
3453:=
3450:1
3418:G
3394:4
3389:C
3353:2
3349:j
3345:=
3340:1
3333:h
3329:j
3326:h
3323:=
3320:)
3317:j
3314:(
3309:h
3297:e
3294:r
3291:e
3288:h
3285:w
3278:,
3275:)
3270:2
3265:C
3255:2
3250:C
3245:(
3239:)
3234:4
3229:C
3212:5
3207:C
3202:(
3180:3
3176:j
3172:h
3169:=
3166:h
3163:j
3141:a
3136:5
3132:/
3125:i
3122:2
3118:e
3114:=
3111:)
3108:a
3105:(
3102:j
3099:=
3096:)
3093:a
3090:(
3087:h
3084:j
3063:a
3058:5
3054:/
3047:i
3044:4
3040:e
3036:=
3033:)
3030:a
3025:5
3021:/
3014:i
3011:2
3007:e
3003:(
3000:h
2997:=
2994:)
2991:a
2988:(
2985:j
2982:h
2953:4
2947:l
2941:0
2935:,
2932:3
2926:n
2920:0
2914:,
2911:1
2905:b
2902:,
2899:a
2893:0
2887:,
2882:l
2878:j
2872:n
2868:h
2862:b
2858:g
2852:a
2848:f
2825:4
2821:j
2815:3
2811:h
2807:g
2804:f
2784:1
2772:.
2759:)
2753:3
2748:+
2743:2
2737:(
2728:5
2724:x
2703:1
2700:=
2695:5
2691:j
2684:,
2681:a
2676:5
2672:/
2665:i
2662:l
2659:2
2655:e
2651:=
2648:)
2645:a
2642:(
2637:l
2633:j
2612:5
2608:/
2603:Z
2595:)
2590:)
2585:5
2581:/
2574:i
2571:2
2567:e
2563:(
2559:)
2554:3
2549:,
2544:2
2539:(
2535:Q
2530:/
2525:)
2521:a
2518:,
2513:5
2509:/
2502:i
2499:2
2495:e
2490:(
2486:)
2481:3
2476:,
2471:2
2466:(
2462:Q
2457:(
2452:t
2449:u
2446:A
2435:.
2423:1
2403:)
2400:1
2394:x
2391:(
2387:/
2383:)
2380:1
2372:5
2368:x
2364:(
2361:=
2358:1
2355:+
2352:x
2349:+
2344:2
2340:x
2336:+
2331:3
2327:x
2323:+
2318:4
2314:x
2293:1
2290:=
2285:4
2281:h
2274:,
2271:3
2265:n
2259:0
2253:,
2248:5
2244:/
2237:i
2234:m
2231:)
2228:1
2225:+
2222:n
2219:(
2216:2
2212:e
2208:=
2204:)
2199:5
2195:/
2188:m
2185:i
2182:2
2178:e
2174:(
2168:n
2164:h
2143:4
2139:/
2134:Z
2126:)
2122:)
2117:3
2112:,
2107:2
2102:(
2098:Q
2093:/
2088:)
2083:5
2079:/
2072:i
2069:2
2065:e
2061:(
2057:)
2052:3
2047:,
2042:2
2037:(
2033:Q
2028:(
2023:t
2020:u
2017:A
2006:.
1994:3
1986:2
1982:x
1961:1
1958:=
1953:2
1949:g
1942:,
1937:3
1929:=
1925:)
1919:3
1910:(
1906:g
1886:2
1882:/
1877:Z
1870:)
1866:)
1861:2
1856:(
1853:Q
1848:/
1844:)
1839:3
1834:,
1829:2
1824:(
1821:Q
1817:(
1812:t
1809:u
1806:A
1795:.
1783:2
1775:2
1771:x
1747:1
1744:=
1739:2
1735:f
1728:,
1723:2
1715:=
1711:)
1705:2
1696:(
1692:f
1672:2
1668:/
1663:Z
1656:)
1652:Q
1647:/
1643:)
1638:2
1633:(
1630:Q
1626:(
1621:t
1618:u
1615:A
1583:)
1579:a
1576:,
1571:5
1567:/
1560:i
1557:2
1553:e
1548:(
1544:)
1539:3
1534:,
1529:2
1524:(
1520:Q
1512:)
1507:5
1503:/
1496:i
1493:2
1489:e
1485:(
1481:)
1476:3
1471:,
1466:2
1461:(
1457:Q
1450:)
1445:3
1440:,
1435:2
1430:(
1426:Q
1419:)
1414:2
1409:(
1405:Q
1397:Q
1367:5
1360:3
1355:+
1350:2
1342:=
1339:a
1315:Q
1286:)
1283:x
1280:(
1277:f
1251:m
1247:F
1223:1
1217:i
1213:F
1206:a
1186:a
1176:i
1172:m
1167:x
1144:i
1117:1
1111:i
1107:F
1096:i
1092:m
1086:i
1061:]
1056:i
1048:[
1043:1
1037:i
1033:F
1029:=
1024:i
1020:F
993:K
990:=
985:m
981:F
966:2
962:F
953:1
949:F
940:0
936:F
932:=
929:F
906:]
903:x
900:[
897:F
891:f
801:e
794:t
787:v
683:8
681:E
675:7
673:E
667:6
665:E
659:4
657:F
651:2
649:G
643:)
641:n
631:)
629:n
619:)
617:n
607:)
605:n
595:)
593:n
583:)
581:n
571:)
569:n
559:)
557:n
499:)
486:Z
474:)
461:Z
437:)
424:Z
415:(
328:p
293:Q
285:n
282:D
272:n
269:A
261:n
258:S
250:n
247:Z
20:)
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