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GroupTheory

  

DirectProduct

  

form the direct product of groups

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

DirectProduct(G1,G2,...)

Parameters

G1,G2, ...

-

group data structures

Description

• 

The DirectProduct command takes a sequence of zero or more groups as input, and returns a group data structure representing the direct product of these groups.

• 

An element of the direct product is a list [s1,s2,...] where s1 is an element from G1, s2 is from G2, and so on.

  

Therefore, the generators defined by DirectProduct are of the form [s1,e2,e3,..], where s1 is a generator from G1, e2 is the identity from G2, e3 is the identity from G3, and so on. Similarly, we have the generators [e1,s2,e3,..],[e1,e2,s3,...] and so forth.

Examples

withGroupTheory:

GDirectProductAlt4,DihedralGroup5,PSL2,3

GA4×D5×PSL2,3

(1)

GroupOrderG

1440

(2)

Use DirectFactor to access the k-th direct factor of a direct product.

DirectFactorG,2

D5

(3)

Access the coordinate projections, as follows.

eRandomElementG

e2,4,3,1,5,4,3,2,2,3,4

(4)

φCanonicalProjectionG,2

φ<a group morphism>

(5)

fφe

f1&comma;5&comma;4&comma;3&comma;2

(6)

finDihedralGroup5

true

(7)

GDirectProductCyclicGroup&comma;Symm3

Gg×S3

(8)

GroupOrderG

(9)

Construct the Cyclic Group of order 2.

GCustomGroup1&comma;`=``.`&comma;a&comma;ba+bmod2&comma;`=``/`&comma;aa

G < a custom group with 1 generator >

(10)

Now form the Klein 4 group.

HDirectProductG&comma;G

H < a custom group with 1 generator > × < a custom group with 1 generator >

(11)

GeneratorsH

1&comma;0&comma;0&comma;1

(12)

We verify the isomorphic permutation form of the Klein 4 group.

AreIsomorphicH&comma;Group1&comma;2&comma;3&comma;4&comma;1&comma;3&comma;2&comma;4

true

(13)

Consider elements of the DirectProduct.

ElementsDirectProductG&comma;G&comma;G

0&comma;0&comma;0&comma;0&comma;0&comma;1&comma;0&comma;1&comma;0&comma;0&comma;1&comma;1&comma;1&comma;0&comma;0&comma;1&comma;0&comma;1&comma;1&comma;1&comma;0&comma;1&comma;1&comma;1

(14)

We verify that DirectProduct is associative and commutative.

K1DirectProductG&comma;DirectProductSymmetricGroup3&comma;CyclicGroup4

K1 < a custom group with 1 generator > ×S3×C4

(15)

K2DirectProductDirectProductG&comma;SymmetricGroup3&comma;CyclicGroup4

K2 < a custom group with 1 generator > ×S3×C4

(16)

AreIsomorphicK1&comma;K2

true

(17)

K1DirectProductH&comma;SymmetricGroup3

K1 < a custom group with 1 generator > × < a custom group with 1 generator > ×S3

(18)

K2DirectProductSymmetricGroup3&comma;H

K2S3× < a custom group with 1 generator > × < a custom group with 1 generator >

(19)

AreIsomorphicK1&comma;K2

true

(20)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[AreIsomorphic]

GroupTheory[CustomGroup]

GroupTheory[CyclicGroup]

GroupTheory[DihedralGroup]

GroupTheory[Elements]

GroupTheory[Group]

GroupTheory[GroupOrder]

GroupTheory[PSL]

GroupTheory[SymmetricGroup]