2022-I  Álgebra Abstracta.

Producto de permutaciones:

G = SymmetricGroup(3)

sigma = G("(1,2,3)")

rho = G("(1,2)")

sigma * rho

Referencia: https://doc.sagemath.org/html/en/thematic_tutorials/group_theory.html#permutation-groups 

Tabla de operaciones de S3:

from sage.matrix.operation_table import OperationTable

G=SymmetricGroup(3)

OperationTable(G, operation=operator.mul)

Tabla de operaciones de Z3:

from sage.matrix.operation_table import OperationTable

R=Integers(6)

OperationTable(R, operation=operator.add)

Tabla de operaciones del grupo alternante A3 con elementos:

from sage.matrix.operation_table import OperationTable

G=AlternatingGroup(3)

OperationTable(G, operator.mul, names='elements')

Referencia: https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/operation_table.html 

Retículo de subgrupos:

G = DihedralGroup(4)

subgroups = G.subgroups()

# Define f(h,k) = True iff h is a subgroup of k

f = lambda h,k: h.is_subgroup(k)

# Define and display the poset

P = Poset((subgroups, f))

P.plot(label_elements=False, vertex_shape= 'H', vertex_size = 800, vertex_colors = 'white')

Referencia: 

• http://sheaves.github.io/Subgroup-Lattice/ 

• https://sheaves.github.io/Subgroup-Lattice-Color-Vertices/ 

Grupos cíclicos (como grupo de permutaciones):

C6 = CyclicPermutationGroup(6)

print("Elementos del grupo", C6.list())

rho = C6("(1,4)(2,5)(3,6)")  

H = C6.subgroup([rho])

print("Elementos del subigrupo", H.list())

Referencia: 

• https://www.math.ucdavis.edu/~anne/SQ2014/thematic_tutorials/group_theory.html 

Grupos Abelianos multiplicativos:

G = AbelianGroup([0,0,0,2,3])

print("Generadores:", G.gens())

print("Ordenes:", G.gens_orders() )

Referencia: 

• https://doc.sagemath.org/html/en/reference/groups/sage/groups/abelian_gps/abelian_group.html 

Grupos Abelianos aditivos:

G = AdditiveAbelianGroup([0, 2, 3]); 

print(G)

print("Generadores:", G.gens())

Referencia: 

• https://doc.sagemath.org/html/en/reference/groups/sage/groups/additive_abelian/additive_abelian_group.html 

Cosets:

G = SymmetricGroup(3) 

a = G("(1,2)") 

H = G.subgroup([a]) 

G.cosets(H, side='right')

Cosets del subgrupo alternante:

G = SymmetricGroup(4)

H = AlternatingGroup(4)

G.cosets(H)