AreIsomorphic - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/AreIsomorphic

GroupTheory

  

AreIsomorphic

  

test if two groups are isomorphic

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

AreIsomorphic(G1, G2)

AreIsomorphic(G1, G2, assign = iso)

iso(g1)

Domain(iso)

Codomain(iso)

Parameters

G1, G2

-

groups

iso

-

mapping returned by AreIsomorphic

g1

-

element of G1

Description

• 

The AreIsomorphic command tests if two groups are isomorphic. It returns true if they are and false if they are not.

• 

If G1 and G2 are indeed isomorphic, then Maple will eventually attempt to construct an isomorphism. You can have this isomorphism assigned to a variable name by using the assign option: if you specify assign=iso, then the isomorphism will be assigned to the variable name iso. This variable can then function as a procedure (or more precisely, a module with ModuleApply) mapping elements from G1 to G2. Concretely, if gG1, and we have specified assign=iso then the call isog will return the element of G2 corresponding to g.

• 

An isomorphism object assigned by AreIsomorphic can be interrogated about its domain and codomain using the Domain and Codomain procedures. If iso was assigned by a call AreIsomorphicG1,G2,assign=iso, then Domainiso returns G1 and Codomainiso returns G2.

Examples

withGroupTheory:

GGL2,3

GGL2,3

(1)

HSmallGroup48,29

H < a permutation group on 48 letters with 5 generators >

(2)

AreIsomorphicH&comma;G

true

(3)

AreIsomorphicG&comma;H&comma;assign=iso

true

(4)

Domainiso

GL2&comma;3

(5)

Codomainiso

< a permutation group on 48 letters with 5 generators >

(6)

aPerm1&comma;6&comma;2&comma;3&comma;4&comma;7&comma;8&comma;5

a1&comma;6&comma;2&comma;34&comma;7&comma;8&comma;5

(7)

bPerm1&comma;2&comma;3&comma;6&comma;4&comma;8&comma;5&comma;7

b1&comma;23&comma;64&comma;85&comma;7

(8)

ainG

true

(9)

binG

true

(10)

a·b

1&comma;3&comma;2&comma;64&comma;5&comma;8&comma;7

(11)

isoa·b

1&comma;7&comma;9&comma;82&comma;14&comma;16&comma;153&comma;19&comma;21&comma;204&comma;24&comma;26&comma;255&comma;28&comma;6&comma;2910&comma;35&comma;37&comma;3611&comma;40&comma;42&comma;4112&comma;44&comma;13&comma;4517&comma;31&comma;18&comma;3222&comma;30&comma;23&comma;2733&comma;47&comma;34&comma;4838&comma;46&comma;39&comma;43

(12)

isoa·isob

1&comma;7&comma;9&comma;82&comma;14&comma;16&comma;153&comma;19&comma;21&comma;204&comma;24&comma;26&comma;255&comma;28&comma;6&comma;2910&comma;35&comma;37&comma;3611&comma;40&comma;42&comma;4112&comma;44&comma;13&comma;4517&comma;31&comma;18&comma;3222&comma;30&comma;23&comma;2733&comma;47&comma;34&comma;4838&comma;46&comma;39&comma;43

(13)

isoa·b

1&comma;7&comma;9&comma;82&comma;14&comma;16&comma;153&comma;19&comma;21&comma;204&comma;24&comma;26&comma;255&comma;28&comma;6&comma;2910&comma;35&comma;37&comma;3611&comma;40&comma;42&comma;4112&comma;44&comma;13&comma;4517&comma;31&comma;18&comma;3222&comma;30&comma;23&comma;2733&comma;47&comma;34&comma;4838&comma;46&comma;39&comma;43

(14)

This example demonstrates that the direct product construction is commutative up to isomorphism.

AAlt4

AA4

(15)

BSymm3

BS3

(16)

GDirectProductA&comma;B

GA4×S3

(17)

HDirectProductB&comma;A

HS3×A4

(18)

AreIsomorphicG&comma;H

true

(19)

Compatibility

• 

The GroupTheory[AreIsomorphic] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[DirectProduct]

GroupTheory[GeneralLinearGroup]

GroupTheory[SmallGroup]

GroupTheory[SymmetricGroup]

Magma[AreIsomorphic]

Perm