Base class for groups

class sage.groups.group.AbelianGroup

Generic abelian group.

static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
is_abelian()
Return True.
class sage.groups.group.AlgebraicGroup

Generic algebraic group.

static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
class sage.groups.group.FiniteGroup

Generic finite group.

static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
cayley_graph()

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

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: 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

Generic group class

__call__()
Coerce x into this group.
__contains__()
x.__contains__(y) <==> y in x
__hash__()
x.__hash__() <==> hash(x)
__init__()
x.__init__(...) initializes x; see x.__class__.__doc__ for signature
static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
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.

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