22.1 Base class for groups

Module: sage.groups.group

Base class for groups

Class: AbelianGroup

class AbelianGroup
Generic abelian group.

Functions: is_abelian

is_abelian( )

Return True.

Class: AlgebraicGroup

class AlgebraicGroup
Generic algebraic group.

Class: FiniteGroup

class FiniteGroup
Generic finite group.

Functions: cayley_graph,$ \,$ is_finite

cayley_graph( )

Returns the cayley graph for this finite group, as a SAGE DiGraph object. To plot the graph with with different colors

sage: D4 = DihedralGroup(4); D4
Dihedral group of order 8 as a permutation group
sage: G = D4.cayley_graph()
sage: show(G, color_by_label=True, edge_labels=True)
sage: A5 = AlternatingGroup(5); A5
Alternating group of order 5!/2 as a permutation group
sage: G = A5.cayley_graph()
sage: G.show3d(color_by_label=True, edge_size=0.01, edge_size2=0.02, vertex_size=0.03)
sage: G.show3d(vertex_size=0.03, edge_size=0.01, edge_size2=0.02, vertex_colors={(1,1,1):G.vertices()}, bgcolor=(0,0,0), color_by_label=True, xres=700, yres=700, iterations=200) # long time (less than a minute)

sage: s1 = SymmetricGroup(1); s = s1.cayley_graph(); s.vertices()
[()]

Author Log:

is_finite( )

Return True.

Class: Group

class Group
Generic group class

Functions: category,$ \,$ is_abelian,$ \,$ is_atomic_repr,$ \,$ is_finite,$ \,$ is_multiplicative,$ \,$ order,$ \,$ quotient,$ \,$ random_element

category( )

The category of all groups

is_abelian( )

Return True if this group is abelian.

is_atomic_repr( )

True if the elements of this group have atomic string representations. For example, integers are atomic but polynomials are not.

is_finite( )

Returns True if this group is finite.

is_multiplicative( )

Returns True if the group operation is given by * (rather than +).

Override for additive groups.

order( )

Returns the number of elements of this group, which is either a positive integer or infinity.

quotient( )

Return the quotient of this group by the normal subgroup $ H$ .

random_element( )

Return a random element of this group.

Special Functions: __call__,$ \,$ __contains__,$ \,$ __init__

__call__( )

Coerce x into this group.

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