GroupTheory/AffineSpecialLinearGroup - Maple Help
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GroupTheory

  

AffineSpecialLinearGroup

  

construct the affine special linear group as a permutation group

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

AffineSpecialLinearGroup( n, q )

ASL( n, q )

Parameters

n

-

a positive integer

q

-

a prime power greater than

Description

• 

The affine special linear group  is the semi-direct product of the special linear group  with the natural module of dimension  over the field with  elements. It is also called the special affine group, and is sometimes denoted by .

• 

The AffineSpecialLinearGroup command produces a permutation group isomorphic to the group .

Examples

Both of the following equivalent commands create a one-dimensional affine special linear group over the field with  elements.

(1)

(2)

(3)

It is clearly a cyclic group of order . In fact, the one-dimensional affine special linear groups are all elementary abelian because, the one-dimensional special linear group being trivial, they are isomorphic to the additive groups of their natural modules.

(4)

(5)

The two-dimensional affine special linear group over a field with  elements is isomorphic to another familiar group.

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

See Also

GroupTheory[AffineGeneralLinearGroup]

GroupTheory[SpecialLinearGroup]

 


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