FreeGroup SΒΆ

free.spad line 425

The free group on a set S is the group of finite products of the form reduce(*, [si ^ ni]) where the si's are in S, and the ni's are integers. The multiplication is not commutative.

1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, S) -> %
x * s returns the product of x by s on the right.
*: (S, %) -> %
s * x returns the product of x by s on the left.
/: (%, %) -> %
from Group
=: (%, %) -> Boolean
from BasicType
^: (%, Integer) -> %
from Group
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
^: (S, Integer) -> %
s ^ n returns the product of s by itself n times.
~=: (%, %) -> Boolean
from BasicType
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: S -> %
from RetractableTo S
commutator: (%, %) -> %
from Group
conjugate: (%, %) -> %
from Group
factors: % -> List Record(gen: S, exp: Integer)
factors(a1\^e1, ..., an\^en) returns [[a1, e1], ..., [an, en]].
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
inv: % -> %
from Group
latex: % -> String
from SetCategory
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mapExpon: (Integer -> Integer, %) -> %
mapExpon(f, a1\^e1 ... an\^en) returns a1\^f(e1) ... an\^f(en).
mapGen: (S -> S, %) -> %
mapGen(f, a1\^e1 ... an\^en) returns f(a1)\^e1 ... f(an)\^en.
nthExpon: (%, Integer) -> Integer
nthExpon(x, n) returns the exponent of the n^th monomial of x.
nthFactor: (%, Integer) -> S
nthFactor(x, n) returns the factor of the n^th monomial of x.
one?: % -> Boolean
from MagmaWithUnit
recip: % -> Union(%, failed)
from MagmaWithUnit
retract: % -> S
from RetractableTo S
retractIfCan: % -> Union(S, failed)
from RetractableTo S
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from MagmaWithUnit
size: % -> NonNegativeInteger
size(x) returns the number of monomials in x.
smaller?: (%, %) -> Boolean if S has Comparable
from Comparable

BasicType

CoercibleTo OutputForm

Comparable if S has Comparable

Group

Magma

MagmaWithUnit

Monoid

RetractableTo S

SemiGroup

SetCategory

unitsKnown