FreeGroup SΒΆ

free.spad line 383 [edit on github]

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 CoercibleFrom S

commutator: (%, %) -> %

from Group

conjugate: (%, %) -> %

from Group

factors: % -> List Record(gen: S, exp: Integer)

factors(a1\^e1, ..., an\^en) returns [[a1, e1], ..., [an, en]].

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

CoercibleFrom S

CoercibleTo OutputForm

Comparable if S has Comparable

Group

Magma

MagmaWithUnit

Monoid

RetractableTo S

SemiGroup

SetCategory

TwoSidedRecip

unitsKnown