RegularWreathProduct - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : RegularWreathProduct

GroupTheory

  

WreathProduct

  

form the wreath product of groups

  

RegularWreathProduct

  

form the regular wreath product of groups

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

WreathProduct( G, H, ... )

RegularWreathProduct( G, H, ... )

Parameters

G,H, ...

-

two or more permutation groups

Description

• 

Let G and H be permutation groups. The wreath product G  H_ of G by H is a permutation group constructed as a semi-direct product of d copies of G (called the base group), where d is the degree of H, and the action of H on the base group is the action of H by permuting the copies of G. Thus, the order of G  H_ is equal to GdH, and the degree of the wreath product is the product of the degrees of G and H.

• 

The regular wreath product of G and H is the wreath product in which H is considered as a regular permutation group on itself. (This is also called the "standard wreath product".)

• 

The WreathProduct( G, H ) command returns a permutation group that is the wreath product G  H_.

• 

If more than two groups are provided as input, then an iterated wreath product is constructed using the left associative rule. For example, WreathProduct( A, B, C, D ) returns ((A  B)  C)  D_.

• 

The RegularWreathProduct( G, H ) command returns the regular wreath product of G and H. In this case, it is not required that H be a permutation group, as a regular permutation representation of the finite group H is used instead. (Here, H may be either a Cayley table group or a finitely presented finite group, as well as a permutation group, which need not be itself regular.)

Examples

withGroupTheory:

GWreathProductCyclicGroup2,CyclicGroup2

G1,2,1,32,4

(1)

AreIsomorphicG,DihedralGroup4

true

(2)

Iterated wreath products appear naturally as Sylow subgroups of symmetric groups of prime power degree.

AreIsomorphicWreathProductCyclicGroup3,CyclicGroup3,SylowSubgroup3,Symm9

true

(3)

AreIsomorphicWreathProductCyclicGroup2$3,SylowSubgroup2,Symm23

true

(4)

Note that iterated wreath products grow quite rapidly.

GroupOrderWreathProductCyclicGroup3,CyclicGroup3

81

(5)

GroupOrderWreathProductCyclicGroup3,CyclicGroup3,CyclicGroup3

1594323

(6)

GroupOrderWreathProductCyclicGroup3,CyclicGroup3,CyclicGroup3,CyclicGroup3

12157665459056928801

(7)

GroupOrderWreathProductCyclicGroup3,CyclicGroup3,CyclicGroup3,CyclicGroup3,CyclicGroup3

5391030899743293631239539488528815119194426882613553319203

(8)

WWreathProductAlt4,Symm3

W < a permutation group on 12 letters with 4 generators >

(9)

GroupOrderW

10368

(10)

The wreath product construction is not commutative; notice that even the order is different.

GroupOrderWreathProductSymm3&comma;Alt4

15552

(11)

Note that the regular wreath product is also different in this case, since the second argument here is not the regular permutation representation of the symmetric group.

GroupOrderRegularWreathProductAlt4&comma;Symm3

17915904

(12)

Since both S3 and A4 are transitive, so too is their wreath product.

IsTransitiveW

true

(13)

However, in general, the wreath product is not primitive.

IsPrimitiveW

false

(14)

BlockSystemW

1&comma;2&comma;3&comma;4&comma;5&comma;6&comma;7&comma;8&comma;9&comma;10&comma;11&comma;12

(15)

Here we construct a wreath product with an intransitive second argument.

WWreathProductAlt4&comma;GroupPerm1&comma;2&comma;3&comma;4&colon;

The resulting group is not transitive.

IsTransitiveW

false

(16)

mapElements&comma;OrbitsW

1&comma;2&comma;3&comma;4&comma;5&comma;6&comma;7&comma;8&comma;9&comma;10&comma;11&comma;12&comma;13&comma;14&comma;15&comma;16

(17)

Here we construct a wreath product with an intransitive first argument.

WWreathProductGroupPerm1&comma;2&comma;3&comma;4&comma;Symm3&colon;

Again, the result is an intransitive group.

IsTransitiveW

false

(18)

mapElements&comma;OrbitsW

1&comma;2&comma;5&comma;6&comma;9&comma;10&comma;3&comma;4&comma;7&comma;8&comma;11&comma;12

(19)

WRegularWreathProductCyclicGroup2&comma;a&comma;b&verbar;a2&comma;b3&comma;`.`a&comma;b5&equals;1

W1&comma;2&comma;1&comma;32&comma;45&comma;136&comma;147&comma;158&comma;169&comma;1710&comma;1811&comma;1912&comma;2021&comma;3722&comma;3823&comma;3924&comma;4025&comma;4126&comma;4227&comma;4328&comma;4429&comma;4530&comma;4631&comma;4732&comma;4833&comma;4934&comma;5035&comma;5136&comma;5253&comma;7354&comma;7455&comma;7556&comma;7657&comma;5958&comma;6061&comma;7762&comma;7863&comma;7964&comma;8065&comma;8166&comma;8267&comma;6968&comma;7071&comma;8372&comma;8485&comma;10786&comma;10887&comma;8988&comma;9091&comma;10992&comma;11093&comma;11194&comma;11295&comma;9796&comma;9899&comma;101100&comma;102103&comma;113104&comma;114105&comma;115106&comma;116117&comma;119118&comma;120&comma;1&comma;5&comma;72&comma;6&comma;83&comma;9&comma;114&comma;10&comma;1213&comma;21&comma;2314&comma;22&comma;2415&comma;25&comma;2716&comma;26&comma;2817&comma;29&comma;3118&comma;30&comma;3219&comma;33&comma;3520&comma;34&comma;3637&comma;51&comma;5338&comma;52&comma;5439&comma;55&comma;5740&comma;56&comma;5841&comma;59&comma;6142&comma;60&comma;6243&comma;63&comma;4544&comma;64&comma;4647&comma;65&comma;6748&comma;66&comma;6849&comma;69&comma;7150&comma;70&comma;7273&comma;85&comma;8774&comma;86&comma;8875&comma;89&comma;9176&comma;90&comma;9277&comma;93&comma;9578&comma;94&comma;9679&comma;97&comma;9980&comma;98&comma;10081&comma;101&comma;10382&comma;102&comma;10483&comma;105&comma;10784&comma;106&comma;108109&comma;117&comma;111110&comma;118&comma;112113&comma;119&comma;115114&comma;120&comma;116

(20)

IsPrimitiveW

false

(21)

BlockSystemW

1&comma;2&comma;3&comma;4&comma;5&comma;6&comma;7&comma;8&comma;9&comma;10&comma;11&comma;12&comma;13&comma;14&comma;15&comma;16&comma;17&comma;18&comma;19&comma;20&comma;21&comma;22&comma;23&comma;24&comma;25&comma;26&comma;27&comma;28&comma;29&comma;30&comma;31&comma;32&comma;33&comma;34&comma;35&comma;36&comma;37&comma;38&comma;39&comma;40&comma;41&comma;42&comma;43&comma;44&comma;45&comma;46&comma;47&comma;48&comma;49&comma;50&comma;51&comma;52&comma;53&comma;54&comma;55&comma;56&comma;57&comma;58&comma;59&comma;60&comma;61&comma;62&comma;63&comma;64&comma;65&comma;66&comma;67&comma;68&comma;69&comma;70&comma;71&comma;72&comma;73&comma;74&comma;75&comma;76&comma;77&comma;78&comma;79&comma;80&comma;81&comma;82&comma;83&comma;84&comma;85&comma;86&comma;87&comma;88&comma;89&comma;90&comma;91&comma;92&comma;93&comma;94&comma;95&comma;96&comma;97&comma;98&comma;99&comma;100&comma;101&comma;102&comma;103&comma;104&comma;105&comma;106&comma;107&comma;108&comma;109&comma;110&comma;111&comma;112&comma;113&comma;114&comma;115&comma;116&comma;117&comma;118&comma;119&comma;120

(22)

SStabilizer109&comma;W&colon;

IndexS&comma;W

120

(23)

GroupOrderS

576460752303423488

(24)

IsElementaryS

true

(25)

DirectFactorsS

< a permutation group on 120 letters with 31 generators > , < a permutation group on 118 letters with 28 generators > , < a permutation group on 116 letters with 27 generators > , < a permutation group on 114 letters with 23 generators > , < a permutation group on 112 letters with 25 generators > , < a permutation group on 108 letters with 28 generators > , < a permutation group on 106 letters with 31 generators > , < a permutation group on 104 letters with 37 generators > , < a permutation group on 102 letters with 31 generators > , < a permutation group on 100 letters with 30 generators > , < a permutation group on 98 letters with 27 generators > , < a permutation group on 96 letters with 32 generators > , < a permutation group on 94 letters with 29 generators > , < a permutation group on 92 letters with 30 generators > , < a permutation group on 90 letters with 23 generators > , < a permutation group on 88 letters with 30 generators > , < a permutation group on 86 letters with 27 generators > , < a permutation group on 84 letters with 25 generators > , < a permutation group on 82 letters with 28 generators > , < a permutation group on 80 letters with 28 generators > , < a permutation group on 78 letters with 35 generators > , < a permutation group on 76 letters with 28 generators > , < a permutation group on 74 letters with 28 generators > , < a permutation group on 72 letters with 27 generators > , < a permutation group on 70 letters with 27 generators > , < a permutation group on 68 letters with 30 generators > , < a permutation group on 66 letters with 20 generators > , < a permutation group on 64 letters with 26 generators > , < a permutation group on 62 letters with 25 generators > , < a permutation group on 60 letters with 29 generators > , < a permutation group on 30 letters with 31 generators > , < a permutation group on 28 letters with 28 generators > , < a permutation group on 26 letters with 34 generators > , < a permutation group on 24 letters with 29 generators > , < a permutation group on 22 letters with 32 generators > , < a permutation group on 20 letters with 32 generators > , < a permutation group on 18 letters with 29 generators > , < a permutation group on 16 letters with 35 generators > , < a permutation group on 14 letters with 30 generators > , < a permutation group on 2 letters with 27 generators > , < a permutation group on 4 letters with 36 generators > , < a permutation group on 6 letters with 29 generators > , < a permutation group on 8 letters with 26 generators > , < a permutation group on 10 letters with 29 generators > , < a permutation group on 12 letters with 32 generators > , < a permutation group on 32 letters with 24 generators > , < a permutation group on 34 letters with 25 generators > , < a permutation group on 36 letters with 25 generators > , < a permutation group on 38 letters with 24 generators > , < a permutation group on 40 letters with 35 generators > , < a permutation group on 42 letters with 29 generators > , < a permutation group on 44 letters with 31 generators > , < a permutation group on 46 letters with 31 generators > , < a permutation group on 48 letters with 28 generators > , < a permutation group on 50 letters with 26 generators > , < a permutation group on 52 letters with 30 generators > , < a permutation group on 54 letters with 28 generators > , < a permutation group on 56 letters with 25 generators > , < a permutation group on 58 letters with 32 generators >

(26)

Notice that, if a group is already regular, then its regular wreath product is isomorphic to the ordinary wreath product.

RTransitiveGroup6&comma;2

R1&comma;3&comma;52&comma;4&comma;6&comma;1&comma;42&comma;35&comma;6

(27)

IsRegularR

true

(28)

AreIsomorphicWreathProductCyclicGroup2&comma;R&comma;RegularWreathProductCyclicGroup2&comma;R

true

(29)

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[AreIsomorphic]

GroupTheory[CyclicGroup]

GroupTheory[DihedralGroup]

GroupTheory[DirectProduct]

GroupTheory[GroupOrder]

GroupTheory[IsPrimitive]

GroupTheory[IsTransitive]

GroupTheory[Orbits]

GroupTheory[SymmetricGroup]