CyclicGroup( n )
CyclicGroup( D, n )
In the first form CyclicGroup
returns the cyclic group of size n as a
permutation group. In the second form D must be a domain of group
elements, e.g., Permutations
or AgWords
, and CyclicGroup
returns
the cyclic group of size n as a group of elements of that type.
gap> c12 := CyclicGroup( 12 ); Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12) ) gap> c105 := CyclicGroup( AgWords, 5*3*7 ); Group( c105_1, c105_2, c105_3 ) gap> Order(c105,c105.1); Order(c105,c105.2); Order(c105,c105.3); 105 35 7
AbelianGroup( sizes )
AbelianGroup( D, sizes )
In the first form AbelianGroup
returns the abelian group C_{sizes[1]}
* C_{sizes[2]} * ... * C_{sizes[n]}, where sizes must be a list
of positive integers, as a permutation group. In the second form D
must be a domain of group elements, e.g., Permutations
or AgWords
,
and AbelianGroup
returns the abelian group as a group of elements of
this type.
gap> g := AbelianGroup( AgWords, [ 2, 3, 7 ] ); Group( a, b, c ) gap> Size( g ); 42 gap> IsAbelian( g ); true
The default function GroupElementsOps.AbelianGroup
uses the functions
CyclicGroup
and DirectProduct
(see DirectProduct) to construct the
abelian group.
ElementaryAbelianGroup( n )
ElementaryAbelianGroup( D, n )
In the first form ElementaryAbelianGroup
returns the elementary abelian
group of size n as a permutation group. n must be a positive prime
power of course. In the second form D must be a domain of group
elements, e.g., Permutations
or AgWords
, and ElementaryAbelianGroup
returns the elementary abelian group as a group of elements of this type.
gap> ElementaryAbelianGroup( 16 ); Group( (1,2), (3,4), (5,6), (7,8) ) gap> ElementaryAbelianGroup( AgWords, 3 ^ 10 ); Group( m59049_1, m59049_2, m59049_3, m59049_4, m59049_5, m59049_6, m59049_7, m59049_8, m59049_9, m59049_10 )
The default function GroupElementsOps.ElementaryAbelianGroup
uses
CyclicGroup
and DirectProduct
(see DirectProduct to construct the
elementary abelian group.
DihedralGroup( n )
DihedralGroup( D, n )
In the first form DihedralGroup
returns the dihedral group of size n
as a permutation group. n must be a positive even integer. In the
second form D must be a domain of group elements, e.g., Permutations
or AgWords
, and DihedralGroup
returns the dihedral group as a group
of elements of this type.
gap> DihedralGroup( 12 ); Group( (1,2,3,4,5,6), (2,6)(3,5) )
PolyhedralGroup( p, q )
PolyhedralGroup( D, p, q )
In the first form PolyhedralGroup
returns the polyhedral group of size
p * q
as a permutation group. p and q must be positive
integers and there must exist a nontrivial p-th root of unity modulo
every prime factor of q. In the second form D must be a domain of
group elements, e.g., Permutations
or Words
, and PolyhedralGroup
returns the polyhedral group as a group of elements of this type.
gap> PolyhedralGroup( 3, 13 ); Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13), ( 2, 4,10)( 3, 7, 6) ( 5,13,11)( 8, 9,12) ) gap> Size( last ); 39
SymmetricGroup( d )
SymmetricGroup( D, d )
In the first form SymmetricGroup
returns the symmetric group of degree
d as a permutation group. d must be a positive integer. In the
second form D must be a domain of group elements, e.g., Permutations
or Words
, and SymmetricGroup
returns the symmetric group as a group
of elements of this type.
gap> SymmetricGroup( 8 ); Group( (1,8), (2,8), (3,8), (4,8), (5,8), (6,8), (7,8) ) gap> Size( last ); 40320
AlternatingGroup( d )
AlternatingGroup( D, d )
In the first form AlternatingGroup
returns the alternating group of
degree d as a permutation group. d must be a positive integer. In
the second form D must be a domain of group elements, e.g.,
Permutations
or Words
, and AlternatingGroup
returns the alternating
group as a group of elements of this type.
gap> AlternatingGroup( 8 ); Group( (1,2,8), (2,3,8), (3,4,8), (4,5,8), (5,6,8), (6,7,8) ) gap> Size( last ); 20160
GeneralLinearGroup( n, q )
GeneralLinearGroup( D, n, q )
In the first form GeneralLinearGroup
returns the general linear group
GL( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
GeneralLinearGroup
returns GL( <n>, <q> ) as a group of elements of
that type.
gap> g := GeneralLinearGroup( 2, 4 ); Size( g ); GL(2,4) 180
SpecialLinearGroup( n, q )
SpecialLinearGroup( D, n, q )
In the first form SpecialLinearGroup
returns the special linear group
SL( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
SpecialLinearGroup
returns SL( <n>, <q> ) as a group of elements of
that type.
gap> g := SpecialLinearGroup( 3, 4 ); Size( g ); SL(3,4) 60480
SymplecticGroup( n, q )
SymplecticGroup( D, n, q )
In the first form SymplecticGroup
returns the symplectic group SP(
<n>, <q> ) as a matrix group. In the second form D must be a domain
of group elements, e.g., Permutations
or AgWords
, and
SymplecticGroup
returns SP( <n>, <q> ) as a group of elements of that
type.
gap> g := SymplecticGroup( 4, 2 ); Size( g ); SP(4,2) 720
GeneralUnitaryGroup( n, q )
GeneralUnitaryGroup( D, n, q )
In the first form GeneralUnitaryGroup
returns the general unitary group
GU( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
GeneralUnitaryGroup
returns GU( <n>, <q> ) as a group of elements of
that type.
gap> g := GeneralUnitaryGroup( 3, 3 ); Size( g ); GU(3,3) 24192
SpecialUnitaryGroup( n, q )
SpecialUnitaryGroup( D, n, q )
In the first form SpecialUnitaryGroup
returns the special unitary group
SU( <n>, <q> ) as a matrix group. In the second form D must be a
domain of group elements, e.g., Permutations
or AgWords
, and
SpecialUnitaryGroup
returns SU( <n>, <q> ) as a group of elements of
that type.
gap> g := SpecialUnitaryGroup( 3, 3 ); Size( g ); SU(3,3) 6048
MathieuGroup( d )
MathieuGroup
returns the Mathieu group of degree d as a permutation
group. d is expected to be 11, 12, 22, 23, or 24.
gap> g := MathieuGroup( 12 ); Size( g ); Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11), ( 3, 7,11, 8) ( 4,10, 5, 6), ( 1,12)( 2,11)( 3, 6)( 4, 8)( 5, 9)( 7,10) ) 95040
GAP 3.4.4