# Automorphisms of abelian groups¶

This implements groups of automorphisms of abelian groups.

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([2,6])
sage: autG = G.aut()


Automorphisms act on the elements of the domain:

sage: g = G.an_element()
sage: f = autG.an_element()
sage: f
Pcgs([ f1, f2, f3 ]) -> [ f1, f1*f2*f3^2, f3^2 ]
sage: (g, f(g))
(f1*f2, f2*f3^2)


Or anything coercible into its domain:

sage: A = AbelianGroup([2,6])
sage: a = A.an_element()
sage: (a, f(a))
(f0*f1, f2*f3^2)
sage: a = A.an_element()
sage: (a, f(a))
((1, 0), f1)
sage: f((1,1))
f2*f3^2


We can compute conjugacy classes:

sage: autG.conjugacy_classes_representatives()
(1,
Pcgs([ f1, f2, f3 ]) -> [ f2*f3, f1*f2, f3 ],
Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2*f3^2, f3^2 ],
[ f3^2, f1*f2*f3, f1 ] -> [ f3^2, f1, f1*f2*f3 ],
Pcgs([ f1, f2, f3 ]) -> [ f2*f3, f1*f2*f3^2, f3^2 ],
[ f1*f2*f3, f1, f3^2 ] -> [ f1*f2*f3, f1, f3 ])


the group order:

sage: autG.order()
12


or create subgroups and do the same for them:

sage: S = autG.subgroup(autG.gens()[:1])
sage: S
Subgroup of automorphisms of Abelian group with gap, generator orders (2, 6)
generated by 1 automorphisms


Only automorphism groups of finite abelian groups are supported:

sage: G = AbelianGroupGap([0,2])        # optional gap_packages
sage: autG = G.aut()                    # optional gap_packages
Traceback (most recent call last):
...
ValueError: only finite abelian groups are supported


AUTHORS:

• Simon Brandhorst (2018-02-17): initial version
class sage.groups.abelian_gps.abelian_aut.AbelianGroupAutomorphism(parent, x, check=True)

Automorphisms of abelian groups with gap.

INPUT:

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([2,3,4,5])
sage: f = G.aut().an_element()

matrix()

Return the matrix defining self.

The $$i$$-th row is the exponent vector of the image of the $$i$$-th generator.

OUTPUT:

• a square matrix over the integers

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([2,3,4])
sage: f = G.aut().an_element()
sage: f
Pcgs([ f1, f2, f3, f4 ]) -> [ f1*f4, f2^2, f1*f3, f4 ]
sage: f.matrix()
[1 0 2]
[0 2 0]
[1 0 1]


Compare with the exponents of the images:

sage: f(G.gens()[0]).exponents()
(1, 0, 2)
sage: f(G.gens()[1]).exponents()
(0, 2, 0)
sage: f(G.gens()[2]).exponents()
(1, 0, 1)

class sage.groups.abelian_gps.abelian_aut.AbelianGroupAutomorphismGroup(AbelianGroupGap)

The full automorphism group of a finite abelian group.

INPUT:

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: from sage.groups.abelian_gps.abelian_aut import AbelianGroupAutomorphismGroup
sage: G = AbelianGroupGap([2,3,4,5])
sage: aut = G.aut()


Equivalently:

sage: aut1 = AbelianGroupAutomorphismGroup(G)
sage: aut is aut1
True

Element
class sage.groups.abelian_gps.abelian_aut.AbelianGroupAutomorphismGroup_gap(domain, gap_group, category, ambient=None)

Base class for groups of automorphisms of abelian groups.

Do not construct this directly.

INPUT:
• domainAbelianGroup_gap
• libgap_parent – the libgap element that is the parent in GAP
• category – a category
• ambient – an instance of a derived class of ParentLibGAP or None (default); the ambient group if libgap_parent has been defined as a subgroup

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: from sage.groups.abelian_gps.abelian_aut import AbelianGroupAutomorphismGroup_gap
sage: domain = AbelianGroupGap([2,3,4,5])
sage: aut = domain.gap().AutomorphismGroupAbelianGroup()
sage: AbelianGroupAutomorphismGroup_gap(domain, aut, Groups().Finite())
<group with 6 generators>

Element
covering_matrix_ring()

Return the covering matrix ring of this group.

This is the ring of $$n \times n$$ matrices over $$\ZZ$$ where $$n$$ is the number of (independent) generators.

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([2,3,4,5])
sage: aut = G.aut()
sage: aut.covering_matrix_ring()
Full MatrixSpace of 4 by 4 dense matrices over Integer Ring

domain()

Return the domain of this group of automorphisms.

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([2,3,4,5])
sage: aut = G.aut()
sage: aut.domain()
Abelian group with gap, generator orders (2, 3, 4, 5)

is_subgroup_of(G)

Return if self is a subgroup of G considered in the same ambient group.

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([2,3,4,5])
sage: aut = G.aut()
sage: gen = aut.gens()
sage: S1 = aut.subgroup(gen[:2])
sage: S1.is_subgroup_of(aut)
True
sage: S2 = aut.subgroup(aut.gens()[1:])
sage: S2.is_subgroup_of(S1)
False

class sage.groups.abelian_gps.abelian_aut.AbelianGroupAutomorphismGroup_subgroup(ambient, generators)

Groups of automorphisms of abelian groups.

They are subgroups of the full automorphism group.

Note

Do not construct this class directly; instead use sage.groups.abelian_gps.abelian_group_gap.AbelianGroup_gap.subgroup().

INPUT:

• ambient – the ambient group
• generators – a tuple of gap elements of the ambient group

EXAMPLES:

sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: from sage.groups.abelian_gps.abelian_aut import AbelianGroupAutomorphismGroup_subgroup
sage: G = AbelianGroupGap([2,3,4,5])
sage: aut = G.aut()
sage: gen = aut.gens()
sage: AbelianGroupAutomorphismGroup_subgroup(aut, gen)
Subgroup of automorphisms of Abelian group with gap, generator orders (2, 3, 4, 5)
generated by 6 automorphisms

Element