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GroupTheory

  

GeneralSemilinearGroup

  

construct a permutation group isomorphic to the General Semi-linear Group over a finite field

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

GeneralSemilinearGroup( n, q )

GammaL( n, q )

Parameters

n

-

a positive integer

q

-

a power of a prime number

Description

• 

The general semi-linear group ΓLn,q is the group of all semi-linear transformations of an n-dimensional vector space over the field with q elements. It is isomorphic to a semi-direct product of the general linear group GLn,q with the Galois group of the field with q elements over its prime sub-field. Thus, if q is prime, then ΓLn,q and GLn,q are isomorphic.

• 

If n and q are positive integers, then the GeneralSemilinearGroup( n, q ) command returns a permutation group isomorphic to the general semi-linear group  ΓLn,q . Otherwise, a symbolic group is returned, for which Maple can do some limited computations.

• 

The abbreviation GammaL( n, q ) is available as a synonym for GeneralSemilinearGroup( n, q ).

• 

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

withGroupTheory:

GGeneralSemilinearGroup2,4

GΓL2,4

(1)

GroupOrderG

360

(2)

GGammaL2,5

GΓL2,5

(3)

GroupOrderG

480

(4)

AreIsomorphicG,GL2,5

true

(5)

GGammaL2,9

GΓL2,9

(6)

csCompositionSeriesG

csΓL2&comma;9 < a permutation group on 80 letters with 4 generators > 1&comma;23&comma;64&comma;85&comma;79&comma;1810&comma;2011&comma;1912&comma;2413&comma;2614&comma;2515&comma;2116&comma;2317&comma;2227&comma;5428&comma;5629&comma;5530&comma;6031&comma;6232&comma;6133&comma;5734&comma;5935&comma;5836&comma;7237&comma;7438&comma;7339&comma;7840&comma;8041&comma;7942&comma;7543&comma;7744&comma;7645&comma;6346&comma;6547&comma;6448&comma;6949&comma;7150&comma;7051&comma;6652&comma;6853&comma;67

(7)

seqGroupOrderS&comma;S&equals;cs

11520,5760,2880,1440,720,2,1

(8)

GroupOrderGammaL4&comma;4

5922201600

(9)

GroupOrderGammaLn&comma;q

logpqk=0n1qnqk

(10)

GroupOrderGammaL3&comma;q

logpqq31q3qq3q2

(11)

See Also

GF

GroupTheory[GeneralLinearGroup]

GroupTheory[GroupOrder]

GroupTheory[SpecialSemilinearGroup]