24.1 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:

(2007-08-10) Bobby Moretti
(2008-05-01) Robert Miller - editing

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 .

random_element( ): Return a random element of this group.

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

__call__( ): Coerce x into this group.

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