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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 1

Description

• 

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

• 

The AffineSpecialLinearGroup command produces a permutation group isomorphic to the group ASLn,q.

Examples

withGroupTheory:

Both of the following equivalent commands create a one-dimensional affine special linear group over the field with 2 elements.

GAffineSpecialLinearGroup1,2

GASL1,2

(1)

GASL1,2

GASL1,2

(2)

GroupOrderG

2

(3)

It is clearly a cyclic group of order 2. 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.

Qselecttype,seq2..100,'primepower':

Gmap2ASL,1,Q:

mapGroupOrder,G

2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49,53,59,61,64,67,71,73,79,81,83,89,97

(4)

andmapIsElementary,G

true

(5)

The two-dimensional affine special linear group over a field with 2 elements is isomorphic to another familiar group.

GASL2,2

GASL2,2

(6)

AreIsomorphicG,Symm4

true

(7)

GASL3,3

GASL3,3

(8)

IsPrimitiveG

true

(9)

TransitivityG

2

(10)

SSocleG

S1,19,102,20,113,21,124,22,135,23,146,24,157,25,168,26,179,27,18,1,3,24,6,57,9,810,12,1113,15,1416,18,1719,21,2022,24,2325,27,26,1,4,72,5,83,6,910,13,1611,14,1712,15,1819,22,2520,23,2621,24,27

(11)

GroupOrderS

27

(12)

IsElementaryS

true

(13)

IsRegularS

true

(14)

See Also

GroupTheory[AffineGeneralLinearGroup]

GroupTheory[SpecialLinearGroup]