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: # needs sage.rings.number_field
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
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)#
Bases:
UniqueRepresentation
,AbelianGroup
Dual of abelian group.
EXAMPLES:
sage: F = AbelianGroup(5,[3,5,7,8,9], names="abcde") sage: F.dual_group() # needs sage.rings.number_field 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
- Element#
alias of
DualAbelianGroupElement
- base_ring()#
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
- gen(i=0)#
The \(i\)-th generator of the abelian group.
EXAMPLES:
sage: # needs sage.rings.number_field 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)
- gens()#
Return the generators for the group.
OUTPUT:
A tuple of group elements generating the group.
EXAMPLES:
sage: F = AbelianGroup([7,11]).dual_group() # needs sage.rings.number_field sage: F.gens() # needs sage.rings.number_field (X0, X1)
- gens_orders()#
The orders of the generators of the dual group.
OUTPUT:
A tuple of integers.
EXAMPLES:
sage: F = AbelianGroup([5]*1000) sage: Fd = F.dual_group() # needs sage.rings.number_field sage: invs = Fd.gens_orders(); len(invs) # needs sage.rings.number_field 1000
- group()#
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
- invariants()#
The invariants of the dual group.
You should use
gens_orders()
instead.EXAMPLES:
sage: F = AbelianGroup([5]*1000) sage: Fd = F.dual_group() # needs sage.rings.number_field sage: invs = Fd.gens_orders(); len(invs) # needs sage.rings.number_field 1000
- is_commutative()#
Return True since this group is commutative.
EXAMPLES:
sage: G = AbelianGroup([2,3,9]) sage: Gd = G.dual_group() # needs sage.rings.number_field sage: Gd.is_commutative() # needs sage.rings.number_field True sage: Gd.is_abelian() # needs sage.rings.number_field True
- list()#
Return tuple of all elements of this group.
EXAMPLES:
sage: G = AbelianGroup([2,3], names="ab") sage: Gd = G.dual_group(names="AB") # needs sage.rings.number_field sage: Gd.list() # needs sage.rings.number_field (1, B, B^2, A, A*B, A*B^2)
- ngens()#
The number of generators of the dual group.
EXAMPLES:
sage: F = AbelianGroup([7]*100) sage: Fd = F.dual_group() # needs sage.rings.number_field sage: Fd.ngens() # needs sage.rings.number_field 100
- order()#
Return the order of this group.
EXAMPLES:
sage: G = AbelianGroup([2,3,9]) sage: Gd = G.dual_group() # needs sage.rings.number_field sage: Gd.order() # needs sage.rings.number_field 54
- random_element()#
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
- sage.groups.abelian_gps.dual_abelian_group.is_DualAbelianGroup(x)#
Return True if \(x\) is the dual group of an abelian group.
EXAMPLES:
sage: # needs sage.rings.number_field 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) 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)