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GroupTheory

  

IsAbelian

  

attempt to determine whether a group is Abelian (commutative)

  

IsCommutative

  

attempt to determine whether a group is commutative

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsAbelian( G )

IsCommutative( G )

Parameters

G

-

a group

Description

• 

A group G is Abelian (or commutative) if every pair of elements of G commute with each other.  That is, for all a and b in G, we have a·b=b·a.

• 

A group is Abelian precisely when it is equal to its own center.

• 

The IsAbelian( G ) command attempts to determine whether the group G is Abelian.  It returns true if G is Abelian and returns false otherwise. The command may raise an exception on (most) finitely presented groups, as shown in the last example below.

• 

The IsCommutative command is provided as an alias.

Examples

withGroupTheory:

GSmallGroup32,1:

IsAbelianG

true

(1)

IsAbelianSmallGroup32,5

false

(2)

Ga|a6=1

Gaa6

(3)

IsAbelianG

true

(4)

IsCommutativea,b,c|a·b=b·a,a·c=c·a,b·c=c·b

true

(5)

IsAbeliana,b|a2,b3,a·b5=1

Error, (in IsAbelian) cannot determine whether a general finitely presented group is Abelian.  If you know that your group is finite, try converting it to a permutation group by using the `PermutationGroup' command with your finitely presented group as input.

Compatibility

• 

The GroupTheory[IsAbelian] and GroupTheory[IsCommutative] commands were introduced in Maple 17.

• 

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

See Also

GroupTheory

GroupTheory[Center]