# PermutationCategory SΒΆ

perm.spad line 1 [edit on github]

S: SetCategory

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

- 1: %
from MagmaWithUnit

- <=: (%, %) -> 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 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

- coerce: % -> OutputForm
from CoercibleTo OutputForm

- commutator: (%, %) -> %
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

- 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

Comparable if S has Finite or S has OrderedSet

OrderedSet if S has Finite or S has OrderedSet

PartialOrder if S has Finite or S has OrderedSet