Base class for groups#
- class sage.groups.group.AbelianGroup#
Bases:
Group
Generic abelian group.
- is_abelian()#
Return True.
EXAMPLES:
sage: from sage.groups.group import AbelianGroup sage: G = AbelianGroup() sage: G.is_abelian() True
- class sage.groups.group.FiniteGroup#
Bases:
Group
Generic finite group.
- is_finite()#
Return
True
.EXAMPLES:
sage: from sage.groups.group import FiniteGroup sage: G = FiniteGroup() sage: G.is_finite() True
- class sage.groups.group.Group#
Bases:
Parent
Base class for all groups
- is_abelian()#
Test whether this group is abelian.
EXAMPLES:
sage: from sage.groups.group import Group sage: G = Group() sage: G.is_abelian() Traceback (most recent call last): ... NotImplementedError
- is_commutative()#
Test whether 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.)
EXAMPLES:
sage: SL(2, 7).is_commutative() # optional - sage.rings.finite_rings False
- is_finite()#
Returns True if this group is finite.
EXAMPLES:
sage: from sage.groups.group import Group sage: G = Group() sage: G.is_finite() Traceback (most recent call last): ... NotImplementedError
- is_multiplicative()#
Returns True if the group operation is given by * (rather than +).
Override for additive groups.
EXAMPLES:
sage: from sage.groups.group import Group sage: G = Group() sage: G.is_multiplicative() True
- order()#
Return the number of elements of this group.
This is either a positive integer or infinity.
EXAMPLES:
sage: from sage.groups.group import Group sage: G = Group() sage: G.order() Traceback (most recent call last): ... NotImplementedError
- quotient(H, **kwds)#
Return the quotient of this group by the normal subgroup \(H\).
EXAMPLES:
sage: from sage.groups.group import Group sage: G = Group() sage: G.quotient(G) Traceback (most recent call last): ... NotImplementedError
- sage.groups.group.is_Group(x)#
Return whether
x
is a group object.INPUT:
x
– anything.
OUTPUT:
Boolean.
EXAMPLES:
sage: F.<a,b> = FreeGroup() sage: from sage.groups.group import is_Group sage: is_Group(F) True sage: is_Group("a string") False