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A<[x]> := AbelianGroup([2,3,4,5,6,10,0]);
> A;
A . 1;
A . 6 ;
Generators(A) ;
NumberOfGenerators(A);
Ngens(A);
Parent(40);
Parent(3);
Relations(A);
RelationMatrix(A) ;
r:= 10=3;
r;
A;
Generators(A) ;
RelationMatrix(A) ;
Ar:=AbelianGroup< A | r > ;
Ar;
Generators(Ar) ;
RelationMatrix(Ar) ;
r[1] ;
r[2] ;
LHS(r);
RHS(r) ;
Parent(r);
UserGenerators(A) ;
UserGenerators(Ar);
As:=sub< A | 3*x[1], 333*x[3], x[5], x[7] >;
As;
Aq:=A/As;
Aq;
Generators(Aq) ;
RelationMatrix(Aq) ;
Invariants(A) ;
Invariants(Ar) ;
Invariants(Aq) ;
TorsionFreeRank(A) ;
TorsionFreeRank(Ar) ;
TorsionFreeRank(Aq) ;
TorsionInvariants(A) ;
TorsionInvariants(Ar) ;
TorsionInvariants(Aq) ;
PrimaryInvariants(A);
PrimaryInvariants(Ar);
PrimaryInvariants(Aq);
pPrimaryInvariants(A, 3);
pPrimaryInvariants(Ar, 2);
pPrimaryInvariants(Aq, 3);
TorsionFreeSubgroup(A) ;
TorsionFreeSubgroup(Ar) ;
TorsionFreeSubgroup(Aq) ;
TorsionSubgroup(A);
TorsionSubgroup(Ar);
TorsionSubgroup(Aq);
pPrimaryComponent(A, 2);
pPrimaryComponent(Ar, 3);
pPrimaryComponent(Aq, 5);
Abelian Group isomorphic to Z/2 + Z/2 + Z/30 + Z/60 + Z
Defined on 7 generators
Relations:
2*x[1] = 0
3*x[2] = 0
4*x[3] = 0
5*x[4] = 0
6*x[5] = 0
10*x[6] = 0
x[1]
x[6]
{
x[3],
x[7],
x[1],
x[2],
x[4],
x[6],
x[5]
}
7
7
Integer Ring
Integer Ring
[ 2*$.1 = 0, 3*$.2 = 0, 4*$.3 = 0, 5*$.4 = 0, 6*$.5 = 0, 10*$.6 = 0 ]
[ 2 0 0 0 0 0 0]
[ 0 3 0 0 0 0 0]
[ 0 0 4 0 0 0 0]
[ 0 0 0 5 0 0 0]
[ 0 0 0 0 6 0 0]
[ 0 0 0 0 0 10 0]
10 = 3
Abelian Group isomorphic to Z/2 + Z/2 + Z/30 + Z/60 + Z
Defined on 7 generators
Relations:
2*x[1] = 0
3*x[2] = 0
4*x[3] = 0
5*x[4] = 0
6*x[5] = 0
10*x[6] = 0
{
x[3],
x[7],
x[1],
x[2],
x[4],
x[6],
x[5]
}
[ 2 0 0 0 0 0 0]
[ 0 3 0 0 0 0 0]
[ 0 0 4 0 0 0 0]
[ 0 0 0 5 0 0 0]
[ 0 0 0 0 6 0 0]
[ 0 0 0 0 0 10 0]
Abelian Group isomorphic to Z/7
Defined on 1 generator
Relations:
7*Ar.1 = 0
{
Ar.1
}
[7]
10
3
10
3
RelationWorld
[
x[1],
x[2],
x[3],
x[4],
x[5],
x[6],
x[7]
]
[
Ar.1
]
Abelian Group isomorphic to Z/2 + Z/2 + Z/12 + Z
Defined on 4 generators in supergroup A:
As.1 = x[1]
As.2 = x[1] + 2*x[3] + 3*x[5]
As.3 = x[1] + x[3] + 5*x[5]
As.4 = x[7]
Relations:
2*As.1 = 0
2*As.2 = 0
12*As.3 = 0
Abelian Group isomorphic to Z/5 + Z/30
Defined on 2 generators
Relations:
5*Aq.1 = 0
30*Aq.2 = 0
{
Aq.2,
Aq.1
}
[ 5 0]
[ 0 30]
[ 2, 2, 30, 60, 0 ]
[ 7 ]
[ 5, 30 ]
1
0
0
[ 2, 2, 30, 60 ]
[ 7 ]
[ 5, 30 ]
[ 2, 2, 2, 4, 3, 3, 5, 5 ]
[ 7 ]
[ 2, 3, 5, 5 ]
[ 3, 3 ]
[]
[ 3 ]
Abelian Group isomorphic to Z
Defined on 1 generator in supergroup A:
$.1 = x[7] (free)
Abelian Group of order 1
Abelian Group of order 1
Abelian Group isomorphic to Z/2 + Z/2 + Z/30 + Z/60
Defined on 4 generators in supergroup A:
$.1 = 2*x[3] + 5*x[6]
$.2 = x[1]
$.3 = x[1] + 4*x[4] + x[5]
$.4 = 2*x[2] + 3*x[3] + x[6]
Relations:
2*$.1 = 0
2*$.2 = 0
30*$.3 = 0
60*$.4 = 0
Abelian Group isomorphic to Z/7
Defined on 1 generator
Relations:
7*Ar.1 = 0
Abelian Group isomorphic to Z/5 + Z/30
Defined on 2 generators
Relations:
5*Aq.1 = 0
30*Aq.2 = 0
Abelian Group isomorphic to Z/2 + Z/2 + Z/2 + Z/4
Defined on 4 generators in supergroup A:
$.1 = 2*x[3] + 5*x[6]
$.2 = x[1]
$.3 = x[1] + 3*x[5]
$.4 = x[3] + 5*x[6]
Relations:
2*$.1 = 0
2*$.2 = 0
2*$.3 = 0
4*$.4 = 0
Abelian Group of order 1
Abelian Group isomorphic to Z/5 + Z/5
Defined on 2 generators in supergroup Aq:
$.1 = Aq.1
$.2 = 6*Aq.2
Relations:
5*$.1 = 0
5*$.2 = 0
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