Semidirect product of groups#

AUTHORS:

  • Mark Shimozono (2013) initial version

class sage.groups.group_semidirect_product.GroupSemidirectProduct(G, H, twist=None, act_to_right=True, prefix0=None, prefix1=None, print_tuple=False, category=Category of groups)[source]#

Bases: CartesianProduct

Return the semidirect product of the groups G and H using the homomorphism twist.

INPUT:

  • G and H – multiplicative groups

  • twist – (default: None) a function defining a homomorphism (see below)

  • act_to_right – True or False (default: True)

  • prefix0 – (default: None) optional string

  • prefix1 – (default: None) optional string

  • print_tuple – True or False (default: False)

  • category – A category (default: Groups())

A semidirect product of groups \(G\) and \(H\) is a group structure on the Cartesian product \(G \times H\) whose product agrees with that of \(G\) on \(G \times 1_H\) and with that of \(H\) on \(1_G \times H\), such that either \(1_G \times H\) or \(G \times 1_H\) is a normal subgroup. In the former case, the group is denoted \(G \ltimes H\) and in the latter, \(G \rtimes H\).

If act_to_right is True, this indicates the group \(G \ltimes H\) in which \(G\) acts on \(H\) by automorphisms. In this case there is a group homomorphism \(\phi \in \mathrm{Hom}(G, \mathrm{Aut}(H))\) such that

\[g h g^{-1} = \phi(g)(h).\]

The homomorphism \(\phi\) is specified by the input twist, which syntactically is the function \(G\times H\to H\) defined by

\[twist(g,h) = \phi(g)(h).\]

The product on \(G \ltimes H\) is defined by

\[\begin{split}\begin{aligned} (g_1,h_1)(g_2,h_2) &= g_1 h_1 g_2 h_2 \\ &= g_1 g_2 g_2^{-1} h_1 g_2 h_2 \\ &= (g_1g_2, twist(g_2^{-1}, h_1) h_2) \end{aligned}\end{split}\]

If act_to_right is False, the group \(G \rtimes H\) is specified by a homomorphism \(\psi\in \mathrm{Hom}(H,\mathrm{Aut}(G))\) such that

\[h g h^{-1} = \psi(h)(g)\]

Then twist is the function \(H\times G\to G\) defined by

\[twist(h,g) = \psi(h)(g).\]

so that the product in \(G \rtimes H\) is defined by

\[\begin{split}\begin{aligned} (g_1,h_1)(g_2,h_2) &= g_1 h_1 g_2 h_2 \\ &= g_1 h_1 g_2 h_1^{-1} h_1 h_2 \\ &= (g_1 twist(h_1,g_2), h_1 h_2) \end{aligned}\end{split}\]

If prefix0 (resp. prefixl) is not None then it is used as a wrapper for printing elements of G (resp. H). If print_tuple is True then elements are printed in the style \((g,h)\) and otherwise in the style \(g * h\).

EXAMPLES:

sage: G = GL(2,QQ)
sage: V = QQ^2
sage: EV = GroupExp()(V)  # make a multiplicative version of V
sage: def twist(g,v):
....:     return EV(g*v.value)
sage: H = GroupSemidirectProduct(G, EV, twist=twist, prefix1='t'); H
Semidirect product of General Linear Group of degree 2
over Rational Field acting on Multiplicative form of Vector space
of dimension 2 over Rational Field
sage: x = H.an_element(); x
t[(1, 0)]
sage: x^2
t[(2, 0)]

sage: # needs sage.rings.number_field
sage: cartan_type = CartanType(['A',2])
sage: W = WeylGroup(cartan_type, prefix="s")
sage: def twist(w,v):
....:     return w*v*(~w)
sage: WW = GroupSemidirectProduct(W, W, twist=twist, print_tuple=True)
sage: s = Family(cartan_type.index_set(), lambda i: W.simple_reflection(i))
sage: y = WW((s[1],s[2])); y
(s1, s2)
sage: y^2
(1, s2*s1)
sage: y.inverse()
(s1, s1*s2*s1)

>>> from sage.all import *
>>> G = GL(Integer(2),QQ)
>>> V = QQ**Integer(2)
>>> EV = GroupExp()(V)  # make a multiplicative version of V
>>> def twist(g,v):
...     return EV(g*v.value)
>>> H = GroupSemidirectProduct(G, EV, twist=twist, prefix1='t'); H
Semidirect product of General Linear Group of degree 2
over Rational Field acting on Multiplicative form of Vector space
of dimension 2 over Rational Field
>>> x = H.an_element(); x
t[(1, 0)]
>>> x**Integer(2)
t[(2, 0)]

>>> # needs sage.rings.number_field
>>> cartan_type = CartanType(['A',Integer(2)])
>>> W = WeylGroup(cartan_type, prefix="s")
>>> def twist(w,v):
...     return w*v*(~w)
>>> WW = GroupSemidirectProduct(W, W, twist=twist, print_tuple=True)
>>> s = Family(cartan_type.index_set(), lambda i: W.simple_reflection(i))
>>> y = WW((s[Integer(1)],s[Integer(2)])); y
(s1, s2)
>>> y**Integer(2)
(1, s2*s1)
>>> y.inverse()
(s1, s1*s2*s1)

Todo

  • Functorial constructor for semidirect products for various categories

  • Twofold Direct product as a special case of semidirect product

Element[source]#

alias of GroupSemidirectProductElement

act_to_right()[source]#

Return True if the left factor acts on the right factor and False if the right factor acts on the left factor.

EXAMPLES:

sage: def twist(x,y):
....:     return y
sage: GroupSemidirectProduct(WeylGroup(['A',2],prefix="s"),
....:                        WeylGroup(['A',3],prefix="t"), twist).act_to_right()
True

>>> from sage.all import *
>>> def twist(x,y):
...     return y
>>> GroupSemidirectProduct(WeylGroup(['A',Integer(2)],prefix="s"),
...                        WeylGroup(['A',Integer(3)],prefix="t"), twist).act_to_right()
True
construction()[source]#

Return None.

This overrides the construction functor inherited from CartesianProduct.

EXAMPLES:

sage: def twist(x,y):
....:     return y
sage: H = GroupSemidirectProduct(WeylGroup(['A',2],prefix="s"),
....:                            WeylGroup(['A',3],prefix="t"), twist)
sage: H.construction()

>>> from sage.all import *
>>> def twist(x,y):
...     return y
>>> H = GroupSemidirectProduct(WeylGroup(['A',Integer(2)],prefix="s"),
...                            WeylGroup(['A',Integer(3)],prefix="t"), twist)
>>> H.construction()
group_generators()[source]#

Return generators of self.

EXAMPLES:

sage: twist = lambda x,y: y
sage: import __main__
sage: __main__.twist = twist
sage: EZ = GroupExp()(ZZ)
sage: GroupSemidirectProduct(EZ, EZ, twist, print_tuple=True).group_generators()
((1, 0), (0, 1))

>>> from sage.all import *
>>> twist = lambda x,y: y
>>> import __main__
>>> __main__.twist = twist
>>> EZ = GroupExp()(ZZ)
>>> GroupSemidirectProduct(EZ, EZ, twist, print_tuple=True).group_generators()
((1, 0), (0, 1))
one()[source]#

The identity element of the semidirect product group.

EXAMPLES:

sage: G = GL(2,QQ)
sage: V = QQ^2
sage: EV = GroupExp()(V)  # make a multiplicative version of V
sage: def twist(g,v):
....:     return EV(g*v.value)
sage: one = GroupSemidirectProduct(G, EV, twist=twist, prefix1='t').one(); one
1
sage: one.cartesian_projection(0)
[1 0]
[0 1]
sage: one.cartesian_projection(1)
(0, 0)

>>> from sage.all import *
>>> G = GL(Integer(2),QQ)
>>> V = QQ**Integer(2)
>>> EV = GroupExp()(V)  # make a multiplicative version of V
>>> def twist(g,v):
...     return EV(g*v.value)
>>> one = GroupSemidirectProduct(G, EV, twist=twist, prefix1='t').one(); one
1
>>> one.cartesian_projection(Integer(0))
[1 0]
[0 1]
>>> one.cartesian_projection(Integer(1))
(0, 0)
opposite_semidirect_product()[source]#

Create the same semidirect product but with the positions of the groups exchanged.

EXAMPLES:

sage: G = GL(2,QQ)
sage: L = QQ^2
sage: EL = GroupExp()(L)
sage: H = GroupSemidirectProduct(G, EL, prefix1='t',
....:                            twist=lambda g,v: EL(g*v.value)); H
Semidirect product of General Linear Group of degree 2
over Rational Field acting on Multiplicative form of Vector space
of dimension 2 over Rational Field
sage: h = H((Matrix([[0,1],[1,0]]), EL.an_element())); h
[0 1]
[1 0] * t[(1, 0)]
sage: Hop = H.opposite_semidirect_product(); Hop
Semidirect product of Multiplicative form of Vector space
of dimension 2 over Rational Field acted upon by
General Linear Group of degree 2 over Rational Field
sage: hop = h.to_opposite(); hop
t[(0, 1)] * [0 1]
[1 0]
sage: hop in Hop
True

>>> from sage.all import *
>>> G = GL(Integer(2),QQ)
>>> L = QQ**Integer(2)
>>> EL = GroupExp()(L)
>>> H = GroupSemidirectProduct(G, EL, prefix1='t',
...                            twist=lambda g,v: EL(g*v.value)); H
Semidirect product of General Linear Group of degree 2
over Rational Field acting on Multiplicative form of Vector space
of dimension 2 over Rational Field
>>> h = H((Matrix([[Integer(0),Integer(1)],[Integer(1),Integer(0)]]), EL.an_element())); h
[0 1]
[1 0] * t[(1, 0)]
>>> Hop = H.opposite_semidirect_product(); Hop
Semidirect product of Multiplicative form of Vector space
of dimension 2 over Rational Field acted upon by
General Linear Group of degree 2 over Rational Field
>>> hop = h.to_opposite(); hop
t[(0, 1)] * [0 1]
[1 0]
>>> hop in Hop
True
product(x, y)[source]#

The product of elements \(x\) and \(y\) in the semidirect product group.

EXAMPLES:

sage: G = GL(2,QQ)
sage: V = QQ^2
sage: EV = GroupExp()(V)  # make a multiplicative version of V
sage: def twist(g,v):
....:     return EV(g*v.value)
sage: S = GroupSemidirectProduct(G, EV, twist=twist, prefix1='t')
sage: g = G([[2,1],[3,1]]); g
[2 1]
[3 1]
sage: v = EV.an_element(); v
(1, 0)
sage: x = S((g,v)); x
[2 1]
[3 1] * t[(1, 0)]
sage: x*x # indirect doctest
[7 3]
[9 4] * t[(0, 3)]

>>> from sage.all import *
>>> G = GL(Integer(2),QQ)
>>> V = QQ**Integer(2)
>>> EV = GroupExp()(V)  # make a multiplicative version of V
>>> def twist(g,v):
...     return EV(g*v.value)
>>> S = GroupSemidirectProduct(G, EV, twist=twist, prefix1='t')
>>> g = G([[Integer(2),Integer(1)],[Integer(3),Integer(1)]]); g
[2 1]
[3 1]
>>> v = EV.an_element(); v
(1, 0)
>>> x = S((g,v)); x
[2 1]
[3 1] * t[(1, 0)]
>>> x*x # indirect doctest
[7 3]
[9 4] * t[(0, 3)]
class sage.groups.group_semidirect_product.GroupSemidirectProductElement[source]#

Bases: Element

Element class for GroupSemidirectProduct.

to_opposite()[source]#

Send an element to its image in the opposite semidirect product.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: L = RootSystem(['A',2]).root_lattice(); L
Root lattice of the Root system of type ['A', 2]
sage: from sage.groups.group_exp import GroupExp
sage: EL = GroupExp()(L)
sage: W = L.weyl_group(prefix="s"); W
Weyl Group of type ['A', 2]
(as a matrix group acting on the root lattice)
sage: def twist(w,v):
....:     return EL(w.action(v.value))
sage: G = GroupSemidirectProduct(W, EL, twist, prefix1='t'); G
Semidirect product of Weyl Group of type ['A', 2] (as a matrix
group acting on the root lattice) acting on Multiplicative form of
Root lattice of the Root system of type ['A', 2]
sage: mu = L.an_element(); mu
2*alpha[1] + 2*alpha[2]
sage: w = W.an_element(); w
s1*s2
sage: g = G((w,EL(mu))); g
s1*s2 * t[2*alpha[1] + 2*alpha[2]]
sage: g.to_opposite()
t[-2*alpha[1]] * s1*s2
sage: g.to_opposite().parent()
Semidirect product of
 Multiplicative form of Root lattice of the Root system of type ['A', 2]
 acted upon by Weyl Group of type ['A', 2]
  (as a matrix group acting on the root lattice)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> L = RootSystem(['A',Integer(2)]).root_lattice(); L
Root lattice of the Root system of type ['A', 2]
>>> from sage.groups.group_exp import GroupExp
>>> EL = GroupExp()(L)
>>> W = L.weyl_group(prefix="s"); W
Weyl Group of type ['A', 2]
(as a matrix group acting on the root lattice)
>>> def twist(w,v):
...     return EL(w.action(v.value))
>>> G = GroupSemidirectProduct(W, EL, twist, prefix1='t'); G
Semidirect product of Weyl Group of type ['A', 2] (as a matrix
group acting on the root lattice) acting on Multiplicative form of
Root lattice of the Root system of type ['A', 2]
>>> mu = L.an_element(); mu
2*alpha[1] + 2*alpha[2]
>>> w = W.an_element(); w
s1*s2
>>> g = G((w,EL(mu))); g
s1*s2 * t[2*alpha[1] + 2*alpha[2]]
>>> g.to_opposite()
t[-2*alpha[1]] * s1*s2
>>> g.to_opposite().parent()
Semidirect product of
 Multiplicative form of Root lattice of the Root system of type ['A', 2]
 acted upon by Weyl Group of type ['A', 2]
  (as a matrix group acting on the root lattice)