# PermutationCategory S¶

PermutationCategory provides a categorical environment for subgroups of bijections of a set (i.e. permutations)

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

/: (%, %) -> %

from Group

<=: (%, %) -> Boolean if S has Finite or S has OrderedSet

from PartialOrder

<: (%, %) -> Boolean

p < q is an order relation on permutations. Note: this order is only total if and only if S is totally ordered or S is finite.

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean if S has Finite or S has OrderedSet

from PartialOrder

>: (%, %) -> Boolean if S has Finite or S has OrderedSet

from PartialOrder

^: (%, Integer) -> %

from Group

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

coerce: % -> OutputForm
commutator: (%, %) -> %

from Group

conjugate: (%, %) -> %

from Group

cycle: List S -> %

cycle(ls) coerces a cycle ls, i.e. a list with not repetitions to a permutation, which maps ls.i to ls.i+1, indices modulo the length of the list. Error: if repetitions occur.

cycles: List List S -> %

cycles(lls) coerces a list list of cycles lls to a permutation, each cycle being a list with not repetitions, is coerced to the permutation, which maps ls.i to ls.i+1, indices modulo the length of the list, then these permutations are mutiplied. Error: if repetitions occur in one cycle.

elt: (%, S) -> S

elt(p, el) returns the image of el under the permutation p.

eval: (%, S) -> S

eval(p, el) returns the image of el under the permutation p.

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

max: (%, %) -> % if S has Finite or S has OrderedSet

from OrderedSet

min: (%, %) -> % if S has Finite or S has OrderedSet

from OrderedSet

one?: % -> Boolean

from MagmaWithUnit

orbit: (%, S) -> Set S

orbit(p, el) returns the orbit of el under the permutation p, i.e. the set which is given by applications of the powers of p to el.

recip: % -> Union(%, failed)

from MagmaWithUnit

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from MagmaWithUnit

smaller?: (%, %) -> Boolean if S has Finite or S has OrderedSet

from Comparable

BasicType

Comparable if S has Finite or S has OrderedSet

Group

Magma

MagmaWithUnit

Monoid

OrderedSet if S has Finite or S has OrderedSet

PartialOrder if S has Finite or S has OrderedSet

SemiGroup

SetCategory

TwoSidedRecip

unitsKnown