GroupTheory
ProjectiveGeneralSemilinearGroup
construct a permutation group isomorphic to the General Semi-linear Group over a finite field
Calling Sequence
Parameters
Description
Examples
ProjectiveGeneralSemilinearGroup( n, q )
PGammaL( n, q )
n
-
a positive integer
q
a power of a prime number
The projective general semi-linear group PΓLn,q is the quotient of the group ΓLn,q of all semi-linear transformations of an n-dimensional vector space over the field with q elements, by the center of GLn,q . 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 PΓLn,q and GLn,q are equal.
If n is a positive integer, and q is a prime power, then the ProjectiveGeneralSemilinearGroup( n, q ) command returns a permutation group isomorphic to the projective general semi-linear group PΓLn,q . Otherwise, a symbolic group is returned, with which Maple can do some limited computations.
The abbreviation PGammaL( n, q ) is available as a synonym for ProjectiveGeneralSemilinearGroup( n, q ).
withGroupTheory:
G≔ProjectiveGeneralSemilinearGroup2,4
G≔2,3,4,1,2,5,3,4
GroupOrderG
120
AreIsomorphicG,Symm5
true
G≔PGammaL2,5
G≔2,4,5,3,1,5,62,3,4
G≔PGammaL2,9
G≔2,7,5,6,3,4,9,8,1,3,104,5,76,8,9,4,85,96,7
1440
ct≔CharacterTableG
Displayct
C
1a
2a
2b
2c
3a
4a
4b
4c
5a
6a
8a
8b
10a
|C|
1
30
36
45
80
90
180
144
240
χ__1
χ__2
−1
χ__3
χ__4
χ__5
9
−3
0
χ__6
χ__7
3
χ__8
χ__9
10
−2
2
χ__10
χ__11
16
−4
χ__12
4
χ__13
20
DrawNormalSubgroupLatticeG
GroupOrderPGammaL3,8
49448448
GroupOrderPGammaLn,q
logpq∏k=0n−1qn−qkq−1
GroupOrderPGammaL5,q
logpqq5−1q5−qq5−q2q5−q3q5−q4q−1
See Also
GF
GroupTheory[GeneralLinearGroup]
GroupTheory[GroupOrder]
GroupTheory[logp]
GroupTheory[SpecialSemilinearGroup]
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