def free(index_set=None, names=None, **kwds):
r"""
Return the free commutative group.
INPUT:
- ``index_set`` -- (optional) an index set for the generators; if
an integer, then this represents `\{0, 1, \ldots, n-1\}`
- ``names`` -- a string or list/tuple/iterable of strings
(default: ``'x'``); the generator names or name prefix
EXAMPLES::
sage: Groups.Commutative.free(index_set=ZZ)
Free abelian group indexed by Integer Ring
sage: Groups().Commutative().free(ZZ)
Free abelian group indexed by Integer Ring
sage: Groups().Commutative().free(5)
Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
sage: F.<x,y,z> = Groups().Commutative().free(); F
Multiplicative Abelian group isomorphic to Z x Z x Z
"""
from sage.rings.all import ZZ
if names is not None:
if isinstance(names, str):
if ',' not in names and index_set in ZZ:
names = [names + repr(i) for i in range(index_set)]
else:
names = names.split(',')
names = tuple(names)
if index_set is None:
index_set = ZZ(len(names))
if index_set in ZZ:
from sage.groups.abelian_gps.abelian_group import AbelianGroup
return AbelianGroup(index_set, names=names, **kwds)
if index_set in ZZ:
from sage.groups.abelian_gps.abelian_group import AbelianGroup
return AbelianGroup(index_set, **kwds)
from sage.groups.indexed_free_group import IndexedFreeAbelianGroup
return IndexedFreeAbelianGroup(index_set, names=names, **kwds)
def FreeGroup(n=None, names='x', index_set=None, abelian=False, **kwds):
"""
Construct a Free Group.
INPUT:
- ``n`` -- integer or ``None`` (default). The number of
generators. If not specified the ``names`` are counted.
- ``names`` -- string or list/tuple/iterable of strings (default:
``'x'``). The generator names or name prefix.
- ``index_set`` -- (optional) an index set for the generators; if
specified then the optional keyword ``abelian`` can be used
- ``abelian`` -- (default: ``False``) whether to construct a free
abelian group or a free group
.. NOTE::
If you want to create a free group, it is currently preferential to
use ``Groups().free(...)`` as that does not load GAP.
EXAMPLES::
sage: from train_track import *
sage: G.<a,b> = FreeGroup(); G
Free Group on generators {a, b}
sage: H = FreeGroup('a, b')
sage: G is H
True
sage: FreeGroup(0)
Free Group on generators {}
The entry can be either a string with the names of the generators,
or the number of generators and the prefix of the names to be
given. The default prefix is ``'x'`` ::
sage: FreeGroup(3)
Free Group on generators {x0, x1, x2}
sage: FreeGroup(3, 'g')
Free Group on generators {g0, g1, g2}
sage: FreeGroup()
Free Group on generators {x}
We give two examples using the ``index_set`` option::
sage: FreeGroup(index_set=ZZ)
Free group indexed by Integer Ring
sage: FreeGroup(index_set=ZZ, abelian=True)
Free abelian group indexed by Integer Ring
TESTS::
sage: G1 = FreeGroup(2, 'a,b')
sage: G2 = FreeGroup('a,b')
sage: G3.<a,b> = FreeGroup()
sage: G1 is G2, G2 is G3
(True, True)
"""
# Support Freegroup('a,b') syntax
if n is not None:
try:
n = Integer(n)
except TypeError:
names = n
n = None
# derive n from counting names
if n is None:
if isinstance(names, six.string_types):
n = len(names.split(','))
else:
names = list(names)
n = len(names)
from sage.structure.category_object import normalize_names
names = normalize_names(n, names)
if index_set is not None or abelian:
if abelian:
from sage.groups.indexed_free_group import IndexedFreeAbelianGroup
return IndexedFreeAbelianGroup(index_set, names=names, **kwds)
from sage.groups.indexed_free_group import IndexedFreeGroup
return IndexedFreeGroup(index_set, names=names, **kwds)
from train_track.free_group import FreeGroup_class
return FreeGroup_class(names)