Dual groups of Finite Multiplicative Abelian Groups

The basic idea is very simple. Let G be an abelian group and \(G^*\) its dual (i.e., the group of homomorphisms from G to \(\CC^\times\)). Let \(g_j\), \(j=1,..,n\), denote generators of \(G\) - say \(g_j\) is of order \(m_j>1\). There are generators \(X_j\), \(j=1,..,n\), of \(G^*\) for which \(X_j(g_j)=\exp(2\pi i/m_j)\) and \(X_i(g_j)=1\) if \(i\not= j\). These are used to construct \(G^*\).

Sage supports multiplicative abelian groups on any prescribed finite number \(n > 0\) of generators. Use AbelianGroup() function to create an abelian group, the dual_group() method to create its dual, and then the gen() and gens() methods to obtain the corresponding generators. You can print the generators as arbitrary strings using the optional names argument to the dual_group() method.

EXAMPLES:

sage: F = AbelianGroup(5, [2,5,7,8,9], names='abcde')
sage: (a, b, c, d, e) = F.gens()

sage: Fd = F.dual_group(names='ABCDE')
sage: Fd.base_ring()
Cyclotomic Field of order 2520 and degree 576
sage: A,B,C,D,E = Fd.gens()
sage: A(a)
-1
sage: A(b), A(c), A(d), A(e)
(1, 1, 1, 1)

sage: # needs sage.rings.real_mpfr
sage: Fd = F.dual_group(names='ABCDE', base_ring=CC)
sage: Fd.category()
Category of commutative groups
sage: A,B,C,D,E = Fd.gens()
sage: A(a)    # abs tol 1e-8
-1.00000000000000 + 0.00000000000000*I
sage: A(b); A(c); A(d); A(e)
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000

>>> from sage.all import *
>>> F = AbelianGroup(Integer(5), [Integer(2),Integer(5),Integer(7),Integer(8),Integer(9)], names='abcde')
>>> (a, b, c, d, e) = F.gens()

>>> Fd = F.dual_group(names='ABCDE')
>>> Fd.base_ring()
Cyclotomic Field of order 2520 and degree 576
>>> A,B,C,D,E = Fd.gens()
>>> A(a)
-1
>>> A(b), A(c), A(d), A(e)
(1, 1, 1, 1)

>>> # needs sage.rings.real_mpfr
>>> Fd = F.dual_group(names='ABCDE', base_ring=CC)
>>> Fd.category()
Category of commutative groups
>>> A,B,C,D,E = Fd.gens()
>>> A(a)    # abs tol 1e-8
-1.00000000000000 + 0.00000000000000*I
>>> A(b); A(c); A(d); A(e)
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000

AUTHORS:

  • David Joyner (2006-08) (based on abelian_groups)

  • David Joyner (2006-10) modifications suggested by William Stein

  • Volker Braun (2012-11) port to new Parent base. Use tuples for immutables. Default to cyclotomic base ring.

class sage.groups.abelian_gps.dual_abelian_group.DualAbelianGroup_class(G, names, base_ring)[source]

Bases: UniqueRepresentation, AbelianGroup

Dual of abelian group.

EXAMPLES:

sage: F = AbelianGroup(5,[3,5,7,8,9], names='abcde')
sage: F.dual_group()
Dual of Abelian Group isomorphic to Z/3Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
over Cyclotomic Field of order 2520 and degree 576

sage: F = AbelianGroup(4,[15,7,8,9], names='abcd')
sage: F.dual_group(base_ring=CC)                                                # needs sage.rings.real_mpfr
Dual of Abelian Group isomorphic to Z/15Z x Z/7Z x Z/8Z x Z/9Z
over Complex Field with 53 bits of precision

>>> from sage.all import *
>>> F = AbelianGroup(Integer(5),[Integer(3),Integer(5),Integer(7),Integer(8),Integer(9)], names='abcde')
>>> F.dual_group()
Dual of Abelian Group isomorphic to Z/3Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
over Cyclotomic Field of order 2520 and degree 576

>>> F = AbelianGroup(Integer(4),[Integer(15),Integer(7),Integer(8),Integer(9)], names='abcd')
>>> F.dual_group(base_ring=CC)                                                # needs sage.rings.real_mpfr
Dual of Abelian Group isomorphic to Z/15Z x Z/7Z x Z/8Z x Z/9Z
over Complex Field with 53 bits of precision
Element[source]

alias of DualAbelianGroupElement

base_ring()[source]

Return the scalars over which the group is dualized.

EXAMPLES:

sage: F = AbelianGroup(3,[5,64,729], names=list("abc"))
sage: Fd = F.dual_group(base_ring=CC)
sage: Fd.base_ring()
Complex Field with 53 bits of precision

>>> from sage.all import *
>>> F = AbelianGroup(Integer(3),[Integer(5),Integer(64),Integer(729)], names=list("abc"))
>>> Fd = F.dual_group(base_ring=CC)
>>> Fd.base_ring()
Complex Field with 53 bits of precision
gen(i=0)[source]

The \(i\)-th generator of the abelian group.

EXAMPLES:

sage: F = AbelianGroup(3, [1,2,3], names='a')
sage: Fd = F.dual_group(names='A')
sage: Fd.0
1
sage: Fd.1
A1
sage: Fd.gens_orders()
(1, 2, 3)

>>> from sage.all import *
>>> F = AbelianGroup(Integer(3), [Integer(1),Integer(2),Integer(3)], names='a')
>>> Fd = F.dual_group(names='A')
>>> Fd.gen(0)
1
>>> Fd.gen(1)
A1
>>> Fd.gens_orders()
(1, 2, 3)
gens()[source]

Return the generators for the group.

OUTPUT: tuple of group elements generating the group

EXAMPLES:

sage: F = AbelianGroup([7,11]).dual_group()
sage: F.gens()
(X0, X1)

>>> from sage.all import *
>>> F = AbelianGroup([Integer(7),Integer(11)]).dual_group()
>>> F.gens()
(X0, X1)
gens_orders()[source]

The orders of the generators of the dual group.

OUTPUT: tuple of integers

EXAMPLES:

sage: F = AbelianGroup([5]*1000)
sage: Fd = F.dual_group()
sage: invs = Fd.gens_orders(); len(invs)
1000

>>> from sage.all import *
>>> F = AbelianGroup([Integer(5)]*Integer(1000))
>>> Fd = F.dual_group()
>>> invs = Fd.gens_orders(); len(invs)
1000
group()[source]

Return the group that self is the dual of.

EXAMPLES:

sage: F = AbelianGroup(3,[5,64,729], names=list("abc"))
sage: Fd = F.dual_group(base_ring=CC)
sage: Fd.group() is F
True

>>> from sage.all import *
>>> F = AbelianGroup(Integer(3),[Integer(5),Integer(64),Integer(729)], names=list("abc"))
>>> Fd = F.dual_group(base_ring=CC)
>>> Fd.group() is F
True
invariants()[source]

The invariants of the dual group.

You should use gens_orders() instead.

EXAMPLES:

sage: F = AbelianGroup([5]*1000)
sage: Fd = F.dual_group()
sage: invs = Fd.gens_orders(); len(invs)
1000

>>> from sage.all import *
>>> F = AbelianGroup([Integer(5)]*Integer(1000))
>>> Fd = F.dual_group()
>>> invs = Fd.gens_orders(); len(invs)
1000
is_commutative()[source]

Return True since this group is commutative.

EXAMPLES:

sage: G = AbelianGroup([2,3,9])
sage: Gd = G.dual_group()
sage: Gd.is_commutative()
True
sage: Gd.is_abelian()
True

>>> from sage.all import *
>>> G = AbelianGroup([Integer(2),Integer(3),Integer(9)])
>>> Gd = G.dual_group()
>>> Gd.is_commutative()
True
>>> Gd.is_abelian()
True
list()[source]

Return a tuple of all elements of this group.

EXAMPLES:

sage: G = AbelianGroup([2,3], names='ab')
sage: Gd = G.dual_group(names='AB')
sage: Gd.list()
(1, B, B^2, A, A*B, A*B^2)

>>> from sage.all import *
>>> G = AbelianGroup([Integer(2),Integer(3)], names='ab')
>>> Gd = G.dual_group(names='AB')
>>> Gd.list()
(1, B, B^2, A, A*B, A*B^2)
ngens()[source]

The number of generators of the dual group.

EXAMPLES:

sage: F = AbelianGroup([7]*100)
sage: Fd = F.dual_group()
sage: Fd.ngens()
100

>>> from sage.all import *
>>> F = AbelianGroup([Integer(7)]*Integer(100))
>>> Fd = F.dual_group()
>>> Fd.ngens()
100
order()[source]

Return the order of this group.

EXAMPLES:

sage: G = AbelianGroup([2,3,9])
sage: Gd = G.dual_group()
sage: Gd.order()
54

>>> from sage.all import *
>>> G = AbelianGroup([Integer(2),Integer(3),Integer(9)])
>>> Gd = G.dual_group()
>>> Gd.order()
54
random_element()[source]

Return a random element of this dual group.

EXAMPLES:

sage: G = AbelianGroup([2,3,9])
sage: Gd = G.dual_group(base_ring=CC)                                       # needs sage.rings.real_mpfr
sage: Gd.random_element().parent() is Gd                                    # needs sage.rings.real_mpfr
True

sage: # needs sage.rings.real_mpfr
sage: N = 43^2 - 1
sage: G = AbelianGroup([N], names='a')
sage: Gd = G.dual_group(names='A', base_ring=CC)
sage: a, = G.gens()
sage: A, = Gd.gens()
sage: x = a^(N/4); y = a^(N/3); z = a^(N/14)
sage: found = [False]*4
sage: while not all(found):
....:     X = A*Gd.random_element()
....:     found[len([b for b in [x,y,z] if abs(X(b)-1)>10^(-8)])] = True

>>> from sage.all import *
>>> G = AbelianGroup([Integer(2),Integer(3),Integer(9)])
>>> Gd = G.dual_group(base_ring=CC)                                       # needs sage.rings.real_mpfr
>>> Gd.random_element().parent() is Gd                                    # needs sage.rings.real_mpfr
True

>>> # needs sage.rings.real_mpfr
>>> N = Integer(43)**Integer(2) - Integer(1)
>>> G = AbelianGroup([N], names='a')
>>> Gd = G.dual_group(names='A', base_ring=CC)
>>> a, = G.gens()
>>> A, = Gd.gens()
>>> x = a**(N/Integer(4)); y = a**(N/Integer(3)); z = a**(N/Integer(14))
>>> found = [False]*Integer(4)
>>> while not all(found):
...     X = A*Gd.random_element()
...     found[len([b for b in [x,y,z] if abs(X(b)-Integer(1))>Integer(10)**(-Integer(8))])] = True
sage.groups.abelian_gps.dual_abelian_group.is_DualAbelianGroup(x)[source]

Return True if \(x\) is the dual group of an abelian group.

EXAMPLES:

sage: from sage.groups.abelian_gps.dual_abelian_group import is_DualAbelianGroup
sage: F = AbelianGroup(5,[3,5,7,8,9], names=list("abcde"))
sage: Fd = F.dual_group()
sage: is_DualAbelianGroup(Fd)
doctest:warning...
DeprecationWarning: the function is_DualAbelianGroup is deprecated;
use 'isinstance(..., DualAbelianGroup_class)' instead
See https://github.com/sagemath/sage/issues/37898 for details.
True
sage: F = AbelianGroup(3,[1,2,3], names='a')
sage: Fd = F.dual_group()
sage: Fd.gens()
(1, X1, X2)
sage: F.gens()
(1, a1, a2)

>>> from sage.all import *
>>> from sage.groups.abelian_gps.dual_abelian_group import is_DualAbelianGroup
>>> F = AbelianGroup(Integer(5),[Integer(3),Integer(5),Integer(7),Integer(8),Integer(9)], names=list("abcde"))
>>> Fd = F.dual_group()
>>> is_DualAbelianGroup(Fd)
doctest:warning...
DeprecationWarning: the function is_DualAbelianGroup is deprecated;
use 'isinstance(..., DualAbelianGroup_class)' instead
See https://github.com/sagemath/sage/issues/37898 for details.
True
>>> F = AbelianGroup(Integer(3),[Integer(1),Integer(2),Integer(3)], names='a')
>>> Fd = F.dual_group()
>>> Fd.gens()
(1, X1, X2)
>>> F.gens()
(1, a1, a2)