Base class for groups

class sage.groups.group.AbelianGroup

Bases: sage.groups.group.Group

Generic abelian group.

is_abelian()

Return True.

class sage.groups.group.AlgebraicGroup

Bases: sage.groups.group.Group

Generic algebraic group.

class sage.groups.group.FiniteGroup

Bases: sage.groups.group.Group

Generic finite group.

cayley_graph(connecting_set=None)

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

INPUT:

`connecting_set` - (optional) list of elements to use for edges,
                   default is the stored generators

EXAMPLES:

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: G.num_edges()
120
sage: G = A5.cayley_graph(connecting_set=[A5.gens()[0]])
sage: G.num_edges()
60
sage: g=PermutationGroup([(i+1,j+1) for i in range(5) for j in range(5) if j!=i])
sage: g.cayley_graph(connecting_set=[(1,2),(2,3)])
Digraph on 120 vertices

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

AUTHORS:

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

is_finite()

Return True.

class sage.groups.group.Group

Bases: sage.structure.parent_gens.ParentWithGens

Generic group class

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_commutative()

Return True if this group is commutative. This is an alias for is_abelian, largely to make groups work well with the Factorization class.

(Note for developers: Derived classes should override is_abelian, not is_commutative.)

EXAMPLE:

sage: SL(2, 7).is_commutative()
False

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(H)

Return the quotient of this group by the normal subgroup .

random_element(bound=None)

Return a random element of this group.