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GroupTheory

  

Center

  

construct the center of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Center( G )

Centre( G )

Parameters

G

-

a permutation group

Description

• 

The center of a group G is the set of elements of G that commute with all elements of G. That is, an element g of G belongs to the center of G if, and only if, g·x=x·g, for all x in G.

• 

The Center( G ) command constructs the center of a group G. The group G must be an instance of a permutation group, a group defined by a Cayley table, or a custom group that defines its own center method.

• 

The Centre command is provided as an alias.

Examples

withGroupTheory:

Whether the center of a dihedral group is trivial or a group of order two depends upon whether the degree is odd or even.

GDihedralGroup6

GD6

(1)

ZCenterG

ZZD6

(2)

GroupOrderZ

2

(3)

GDihedralGroup7

GD7

(4)

ZCenterG

ZZD7

(5)

GroupOrderZ

1

(6)

CenterAlternatingGroup4

(7)

GGL3,3

GGL3,3

(8)

IsAbelianCenterG

true

(9)

GroupOrderCenterG

2

(10)

IsNormalCenterG,G

true

(11)

The center of any Frobenius group is trivial.

GFrobeniusGroup72,2

G2,8,4,73,9,6,5,2,3,4,65,7,9,8,2,43,65,97,8,1,2,43,5,76,8,9,1,3,62,5,84,7,9

(12)

GroupOrderCentreG

1

(13)

Likewise, a non-abelian simple group has trivial center.

CentreMcLaughlinGroup

(14)

Of course, every abelian group is equal to its center.

CentreCyclicGroup24

C24

(15)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[Centralizer]