def group_law(G):
r"""
Return a triple ``(identity, operation, inverse)`` that define the
operations on the group ``G``.
EXAMPLES::
sage: from sage.combinat.designs.difference_family import group_law
sage: group_law(Zmod(3))
(0, <built-in function add>, <built-in function neg>)
sage: group_law(SymmetricGroup(5))
((), <built-in function mul>, <built-in function inv>)
sage: group_law(VectorSpace(QQ,3))
((0, 0, 0), <built-in function add>, <built-in function neg>)
"""
import operator
from sage.categories.groups import Groups
from sage.categories.additive_groups import AdditiveGroups
if G in Groups(): # multiplicative groups
return (G.one(), operator.mul, operator.inv)
elif G in AdditiveGroups(): # additive groups
return (G.zero(), operator.add, operator.neg)
else:
raise ValueError("%s does not seem to be a group" % G)
def _determine_category_(category):
r"""
Return the category of this imaginary group.
INPUT:
- ``category`` -- a category or ``None`` (in which case the output
equals ``category``)
OUTPUT:
A category.
EXAMPLES::
sage: from sage.groups.misc_gps.imaginary_groups import ImaginaryGroup
sage: from sage.categories.additive_groups import AdditiveGroups
sage: ImaginaryGroup._determine_category_(None)
Category of commutative additive groups
sage: ImaginaryGroup._determine_category_(AdditiveGroups())
Category of additive groups
"""
if category is None:
from sage.categories.additive_groups import AdditiveGroups
category = AdditiveGroups().AdditiveCommutative()
return category