# PermutationCategory SΒΆ

- S: SetCategory

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

- <: (%, %) -> 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
- 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
- from CoercibleTo 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

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