# Permutation groups

## Constructing permutation groups

Permutation groups can be defined as symmetric groups, alternating groups or their subgroups.

`PermGroup`

— Type`PermGroup`

Groups of permutations. Every group of this type is a subgroup of Sym(n) for some n.

**Examples**

`symmetric_group(n::Int)`

: the symmetric group Sym(n)`alternating_group(n::Int)`

: the alternating group Alt(n)- subgroups of Sym(n)
`dihedral_group(PermGroup, n::Int)`

: the dihedral group of order`n`

as a group of permutations. Same holds replacing`dihedral_group`

by`quaternion_group`

If `G`

is a permutation group and `x`

is a permutation, `G(x)`

returns a permutation `x`

with parent `G`

; an exception is thrown if `x`

does not embed into `G`

.

```
julia> G=symmetric_group(5)
Sym(5)
julia> x=cperm([1,2,3])
(1,2,3)
julia> parent(x)
Sym(3)
julia> y=G(x)
(1,2,3)
julia> parent(y)
Sym(5)
```

If `G`

is a permutation group and `x`

is a vector of integers, `G(x)`

returns a `PermGroupElem`

with parent `G`

; an exception is thrown if the element does not embed into `G`

.

**Examples**

```
julia> G = symmetric_group(6)
Sym(6)
julia> x = G([2,4,6,1,3,5])
(1,2,4)(3,6,5)
julia> parent(x)
Sym(6)
```

`PermGroupElem`

— Type`PermGroupElem`

Element of a group of permutations. It is displayed as product of disjoint cycles.

**Assumptions:**

- for
`x`

,`y`

in Sym(n), the product`xy`

is read from left to right; - for
`x`

in Sym(n) and`i`

in {1,...,n},`i^x`

and`x(i)`

return the image of`i`

under the action of`x`

.

`symmetric_group`

— Function`symmetric_group(n::Int)`

Return the full symmetric group on the set `{1, 2, ..., n}`

.

**Examples**

```
julia> G = symmetric_group(5)
Sym(5)
julia> order(G)
120
```

`alternating_group`

— Function`alternating_group(n::Int)`

Return the full alternating group on the set `{1, 2, ..., n}`

..

**Examples**

```
julia> G = alternating_group(5)
Alt(5)
julia> order(G)
60
```

`permutation_group`

— Function`permutation_group(n::IntegerUnion, perms::Vector{PermGroupElem})`

Return the permutation group of degree `n`

that is generated by the elements in `perms`

.

**Examples**

```
julia> x = cperm([1,2,3], [4,5]); y = cperm([1,4]);
julia> permutation_group(5, [x, y])
Permutation group of degree 5
```

`@permutation_group`

— Macro`@permutation_group(n, gens...)`

Input the permutation group of degree `n`

with generators `gens...`

, given by permutations in cycle notation.

**Examples**

```
julia> g = @permutation_group(7, (1,2), (1,2,3)(4,5))
Permutation group of degree 7
julia> degree(g)
7
```

`projective_general_linear_group`

— Function`projective_general_linear_group(n::Int, q::Int)`

Return the factor group of `general_linear_group`

, called with the same parameters, by its scalar matrices. The group is represented as a permutation group.

**Examples**

```
julia> g = projective_general_linear_group(2, 3)
Permutation group of degree 4 and order 24
julia> order(g)
24
```

`projective_special_linear_group`

— Function`projective_special_linear_group(n::Int, q::Int)`

Return the factor group of `special_linear_group`

, called with the same parameters, by its scalar matrices. The group is represented as a permutation group.

**Examples**

```
julia> g = projective_special_linear_group(2, 3)
Permutation group of degree 4 and order 12
julia> order(g)
12
```

`projective_symplectic_group`

— Function`projective_symplectic_group(n::Int, q::Int)`

Return the factor group of `symplectic_group`

, called with the same parameters, by its scalar matrices. The group is represented as a permutation group.

**Examples**

```
julia> g = projective_symplectic_group(2, 3)
Permutation group of degree 4 and order 12
julia> order(g)
12
```

`projective_orthogonal_group`

— Function`projective_orthogonal_group(e::Int, n::Int, q::Int)`

Return the factor group of `orthogonal_group`

, called with the same parameters, by its scalar matrices.

As for `orthogonal_group`

, `e`

can be omitted if `n`

is odd.

**Examples**

```
julia> g = projective_orthogonal_group(1, 4, 3); order(g)
576
julia> g = projective_orthogonal_group(3, 3); order(g)
24
```

`projective_special_orthogonal_group`

— Function`projective_special_orthogonal_group(e::Int, n::Int, q::Int)`

Return the factor group of `special_orthogonal_group`

, called with the same parameters, by its scalar matrices.

As for `special_orthogonal_group`

, `e`

can be omitted if `n`

is odd.

**Examples**

```
julia> g = projective_special_orthogonal_group(1, 4, 3); order(g)
288
julia> g = projective_special_orthogonal_group(3, 3); order(g)
24
```

`projective_omega_group`

— Function`projective_omega_group(e::Int, n::Int, q::Int)`

Return the factor group of `omega_group`

, called with the same parameters, by its scalar matrices.

As for `omega_group`

, `e`

can be omitted if `n`

is odd.

**Examples**

```
julia> g = projective_omega_group(1, 4, 3); order(g)
144
julia> g = projective_omega_group(3, 3); order(g)
12
```

`projective_unitary_group`

— Function`projective_unitary_group(n::Int, q::Int)`

Return the factor group of `unitary_group`

, called with the same parameters, by its scalar matrices. The group is represented as a permutation group.

**Examples**

```
julia> g = projective_unitary_group(2, 3)
Permutation group of degree 10 and order 24
julia> order(g)
24
```

`projective_special_unitary_group`

— Function`projective_special_unitary_group(n::Int, q::Int)`

Return the factor group of `special_unitary_group`

, called with the same parameters, by its scalar matrices. The group is represented as a permutation group.

**Examples**

```
julia> g = projective_special_unitary_group(2, 3)
Permutation group of degree 10 and order 12
julia> order(g)
12
```

## Operations for permutation groups

All operations, properties and attributes for general groups described in the previous sections are supported for permutation groups. In addition there are some specific to permutation groups.

In OSCAR, every permutation group has a degree `n`

, that corresponds to the size of the set on which `G`

acts.

`degree`

— Method`degree(G::PermGroup) -> Int`

Return the degree of `G`

as a permutation group, that is, an integer `n`

that is stored in `G`

, with the following meaning.

`G`

embeds into`symmetric_group(n)`

.- Two permutation groups of different degrees are regarded as not equal, even if they contain the same permutations.
- Subgroups constructed with
`derived_subgroup`

,`sylow_subgroup`

, etc., get the same degree as the given group. - The range
`1:degree(G)`

is used as the default set of points on which`G`

and its element acts.

The degree of a group of permutations is not necessarily equal to the largest moved point of the group `G`

. For example, the trivial subgroup of `symmetric_group(n)`

has degree `n`

even though it fixes `n`

.

**Examples**

```
julia> degree(symmetric_group(4))
4
julia> t4 = trivial_subgroup(symmetric_group(4))[1];
julia> degree(t4)
4
julia> t4 == trivial_subgroup(symmetric_group(5))[1]
false
julia> show(Vector(gen(symmetric_group(4), 2)))
[2, 1, 3, 4]
julia> show(Vector(gen(symmetric_group(5), 2)))
[2, 1, 3, 4, 5]
```

The following functions deal with the natural action of a given permutation group $G$.

`is_transitive`

— Function`is_transitive(G::PermGroup, L::AbstractVector{Int} = 1:degree(G))`

Return whether `G`

acts transitively on `L`

, that is, `L`

is an orbit of `G`

.

**Examples**

```
julia> G = symmetric_group(6);
julia> is_transitive(G)
true
julia> is_transitive(sylow_subgroup(G, 2)[1])
false
julia> is_transitive(stabilizer(G, 1)[1])
false
```

`transitivity`

— Function`transitivity(G::PermGroup, L::AbstractVector{Int} = 1:degree(G))`

Return the maximum `k`

such that `G`

acts `k`

-transitively on `L`

, that is, every `k`

-tuple of points in `L`

can be mapped simultaneously to every other `k`

-tuple by an element of `G`

.

The output is `0`

if `G`

acts intransitively on `L`

, and an exception is thrown if `G`

does not act on `L`

.

**Examples**

```
julia> transitivity(mathieu_group(24))
5
julia> transitivity(symmetric_group(6))
6
julia> transitivity(symmetric_group(6), 1:7)
0
julia> transitivity(symmetric_group(6), 1:5)
ERROR: ArgumentError: the group does not act
```

`is_primitive`

— Function`is_primitive(G::PermGroup, L::AbstractVector{Int} = 1:degree(G))`

Return whether the action of `G`

on `L`

is primitive, that is, the action is transitive and the point stabilizers are maximal in `G`

.

**Examples**

```
julia> G = alternating_group(6);
julia> mx = filter(is_transitive, map(representative, maximal_subgroup_classes(G)))
3-element Vector{PermGroup}:
Permutation group of degree 6 and order 24
Permutation group of degree 6 and order 36
Permutation group of degree 6 and order 60
julia> [(order(H), is_primitive(H)) for H in mx]
3-element Vector{Tuple{ZZRingElem, Bool}}:
(24, 0)
(36, 0)
(60, 1)
```

`is_regular`

— Function`is_regular(G::PermGroup, L::AbstractVector{Int} = 1:degree(G))`

Return whether the action of `G`

on `L`

is regular (i.e., transitive and semiregular).

**Examples**

```
julia> G = symmetric_group(6);
julia> H = sub(G, [G([2, 3, 4, 5, 6, 1])])[1]
Permutation group of degree 6
julia> is_regular(H)
true
julia> is_regular(G)
false
```

`is_semiregular`

— Function`is_semiregular(G::PermGroup, L::AbstractVector{Int} = 1:degree(G))`

Return whether the action of `G`

on `L`

is semiregular (i.e., the stabilizer of each point is the identity).

**Examples**

```
julia> G = symmetric_group(6);
julia> H = sub(G, [G([2, 3, 1, 5, 6, 4])])[1]
Permutation group of degree 6
julia> is_semiregular(H)
true
julia> is_regular(H)
false
```

`rank_action`

— Function`rank_action(G::PermGroup, L::AbstractVector{Int} = 1:degree(G))`

Return the rank of the transitive action of `G`

on `L`

. This is defined as the number of `G`

-orbits in the action on ordered pairs of points in `L`

, and is equal to the number of orbits of the stabilizer of a point in `L`

on `L`

, see [Cam99] Section 1.11.

An exception is thrown if `G`

is not transitive on `L`

.

**Examples**

```
julia> G = symmetric_group(4); rank_action(G) # 4-transitive
2
julia> H = sylow_subgroup(G, 2)[1]
Permutation group of degree 4 and order 8
julia> rank_action(H) # not 2-transitive
3
julia> K = stabilizer(G, 1)[1]
Permutation group of degree 4 and order 6
julia> rank_action(K, 2:4) # 2-transitive
2
julia> rank_action(K, 3:5)
ERROR: ArgumentError: the group is not transitive
```

`blocks`

— Function`blocks(G::PermGroup, L::AbstractVector{Int} = moved_points(G))`

Return a G-set that is a block system for the action of `G`

on `L`

, i.e., a non-trivial partition of `L`

preserved by the action of `G`

.

Here, `L`

must be a subvector of `1:degree(G)`

on which `G`

acts transitively. `G`

may move points outside `L`

, in this case the restriction of the action of the set stabilizer of `L`

in `G`

to `L`

is considered.

An exception is thrown if this action is not transitive.

**Examples**

```
julia> g = sylow_subgroup(symmetric_group(4), 2)[1]
Permutation group of degree 4 and order 8
julia> collect(blocks(g))
2-element Vector{Vector{Int64}}:
[1, 2]
[3, 4]
```

`maximal_blocks`

— Function`maximal_blocks(G::PermGroup, L::AbstractVector{Int} = moved_points(G))`

Return a G-set that is a maximal block system for the action of `G`

on `L`

, i.e., a maximal non-trivial partition of `L`

preserved by the action of `G`

.

Here, `L`

must be a subvector of `1:degree(G)`

on which `G`

acts transitively. `G`

may move points outside `L`

, in this case the restriction of the action of the set stabilizer of `L`

in `G`

to `L`

is considered.

An exception is thrown if this action is not transitive.

**Examples**

```
julia> G = transitive_group(8, 2)
Permutation group of degree 8
julia> collect(maximal_blocks(G))
2-element Vector{Vector{Int64}}:
[1, 2, 3, 8]
[4, 5, 6, 7]
```

`minimal_block_reps`

— Function`minimal_block_reps(G::PermGroup, L::AbstractVector{Int} = moved_points(G))`

Return a vector of block representatives for all minimal non-trivial block systems for the action of `G`

on `L`

.

Here, `L`

must be a subvector of `1:degree(G)`

on which `G`

acts transitively. `G`

may move points outside `L`

, in this case the restriction of the action of the set stabilizer of `L`

in `G`

to `L`

is considered.

An exception is thrown if this action is not transitive.

**Examples**

```
julia> G = transitive_group(8, 2)
Permutation group of degree 8
julia> minimal_block_reps(G)
3-element Vector{Vector{Int64}}:
[1, 3]
[1, 5]
[1, 7]
```

`all_blocks`

— Method`all_blocks(G::PermGroup)`

Return a vector of smallest representatives of all block systems for the action of `G`

on the set of moved points of `G`

.

**Examples**

```
julia> G = transitive_group(8, 2)
Permutation group of degree 8
julia> all_blocks(G)
6-element Vector{Vector{Int64}}:
[1, 2, 3, 8]
[1, 5]
[1, 3, 5, 7]
[1, 3]
[1, 3, 4, 6]
[1, 7]
```

The following functions allow efficiently "recognizing" certain permutation groups as alternating or symmetric groups.

`is_natural_symmetric_group`

— Method`is_natural_symmetric_group(G::GAPGroup)`

Return `true`

if `G`

is a permutation group acting as the symmetric group on its moved points, and `false`

otherwise.

`is_isomorphic_to_symmetric_group`

— Method`is_isomorphic_to_symmetric_group(G::GAPGroup)`

Return `true`

if `G`

is isomorphic to a symmetric group, and `false`

otherwise.

`is_natural_alternating_group`

— Method`is_natural_alternating_group(G::GAPGroup)`

Return `true`

if `G`

is a permutation group acting as the alternating group on its moved points, and `false`

otherwise.

`is_isomorphic_to_alternating_group`

— Method`is_isomorphic_to_alternating_group(G::GAPGroup)`

Return `true`

if `G`

is isomorphic to an alternating group, and `false`

otherwise.

## Permutations

Permutations in OSCAR are displayed as products of disjoint cycles, as in GAP. An explicit permutation can be built using the functions `perm`

, `cperm`

, or `@perm`

.

`perm`

— Function`perm(L::AbstractVector{<:IntegerUnion})`

Return the permutation $x$ which maps every $i$ from `1`

to $n$`= length(L)`

to `L`

$[i]$. The parent of $x$ is set to `symmetric_group`

$(n)$. An exception is thrown if `L`

does not contain every integer from 1 to $n$ exactly once.

The parent group of $x$ is set to `symmetric_group`

$(n)$.

**Examples**

```
julia> x = perm([2,4,6,1,3,5])
(1,2,4)(3,6,5)
julia> parent(x)
Sym(6)
```

```
perm(G::PermGroup, L::AbstractVector{<:IntegerUnion})
(G::PermGroup)(L::AbstractVector{<:IntegerUnion})
```

Return the permutation $x$ which maps every `i`

from 1 to $n$`= length(L)`

to `L`

$[i]$. The parent of $x$ is `G`

. An exception is thrown if $x$ is not contained in `G`

or `L`

does not contain every integer from 1 to $n$ exactly once.

**Examples**

```
julia> perm(symmetric_group(6),[2,4,6,1,3,5])
(1,2,4)(3,6,5)
```

Equivalent permutations can be created using `cperm`

and `@perm`

```
julia> x = perm(symmetric_group(8),[2,3,1,5,4,7,8,6])
(1,2,3)(4,5)(6,7,8)
julia> y = cperm([1,2,3],[4,5],[6,7,8])
(1,2,3)(4,5)(6,7,8)
julia> x == y
true
julia> z = @perm (1,2,3)(4,5)(6,7,8)
(1,2,3)(4,5)(6,7,8)
julia> x == z
true
```

`cperm`

— Function```
cperm(L::AbstractVector{<:T}...) where T <: IntegerUnion
cperm(G::PermGroup, L::AbstractVector{<:T}...)
cperm(L::Vector{Vector{T}}) where T <: IntegerUnion
cperm(g::PermGroup,L::Vector{Vector{T}}) where T <: IntegerUnion
```

For given lists $[a_1, a_2, \ldots, a_n], [b_1, b_2, \ldots , b_m], \ldots$ of positive integers, return the permutation $x = (a_1, a_2, \ldots, a_n) * (b_1, b_2, \ldots, b_m) * \ldots$. Arrays of the form `[n, n+1, ..., n+k]`

can be replaced by `n:n+k`

.

The parent of $x$ is `G`

. If `G`

is not specified then the parent of $x$ is set to `symmetric_group`

$(n)$, where $n$ is the largest integer that occurs in an entry of `L`

.

An exception is thrown if $x$ is not contained in `G`

or one of the given vectors is empty or contains duplicates.

**Examples**

```
julia> cperm([1,2,3],4:7)
(1,2,3)(4,5,6,7)
julia> cperm([1,2],[2,3])
(1,3,2)
julia> cperm()
()
julia> p = cperm([1,2,3],[7])
(1,2,3)
julia> degree(p)
7
```

Two permutations coincide if, and only if, they move the same points and their parent groups have the same degree.

```
julia> G=symmetric_group(5);
julia> A=alternating_group(5);
julia> x=cperm(G,[1,2,3]);
julia> y=cperm(A,[1,2,3]);
julia> z=cperm([1,2,3]); parent(z)
Sym(3)
julia> x==y
true
julia> x==z
false
```

In the example above, `x`

and `y`

are equal because both act on a set of cardinality `5`

, while `x`

and `z`

are different because `x`

belongs to `Sym(5)`

and `z`

belongs to `Sym(3)`

.

cperm can also handle cycles passed in inside of a vector

```
julia> x = cperm([[1,2],[3,4]])
(1,2)(3,4)
julia> y = cperm([1,2],[3,4])
(1,2)(3,4)
julia> x == y
true
```

```
julia> G=symmetric_group(5)
Sym(5)
julia> x = cperm(G,[[1,2],[3,4]])
(1,2)(3,4)
julia> parent(x)
Sym(5)
```

Equivalent permutations can be created using `perm`

and `@perm`

:

```
julia> x = cperm([1,2,3],[4,5],[6,7,8])
(1,2,3)(4,5)(6,7,8)
julia> y = perm(symmetric_group(8),[2,3,1,5,4,7,8,6])
(1,2,3)(4,5)(6,7,8)
julia> x == y
true
julia> z = @perm (1,2,3)(4,5)(6,7,8)
(1,2,3)(4,5)(6,7,8)
julia> x == z
true
```

At the moment, the input vectors of the function `cperm`

need not be disjoint.

`@perm`

— Macro`@perm ex`

Input a permutation in cycle notation. Supports arbitrary expressions for generating the integer entries of the cycles. The parent group is inferred to be the symmetric group with a degree of the highest integer referenced in the permutation.

The actual work is done by `cperm`

. Thus, for the time being, cycles which are *not* disjoint actually are supported.

**Examples**

```
julia> x = @perm (1,2,3)(4,5)(factorial(3),7,8)
(1,2,3)(4,5)(6,7,8)
julia> parent(x)
Sym(8)
julia> y = cperm([1,2,3],[4,5],[6,7,8])
(1,2,3)(4,5)(6,7,8)
julia> x == y
true
julia> z = perm(symmetric_group(8),[2,3,1,5,4,7,8,6])
(1,2,3)(4,5)(6,7,8)
julia> x == z
true
```

`@perm n gens`

Input a list of permutations in cycle notation, created as elements of the symmetric group of degree `n`

, i.e., `symmetric_group(n)`

, by invoking `cperm`

suitably.

**Examples**

```
julia> gens = @perm 14 [
(1,10)
(2,11)
(3,12)
(4,13)
(5,14)
(6,8)
(7,9)
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)
(1,2)(10,11)
]
9-element Vector{PermGroupElem}:
(1,10)
(2,11)
(3,12)
(4,13)
(5,14)
(6,8)
(7,9)
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)
(1,2)(10,11)
julia> parent(gens[1])
Sym(14)
```

The function `Vector{T}`

works in the opposite way with respect to `perm`

:

`Vector`

— Type```
Vector{T}(x::PermGroupElem, n::Int = x.parent.deg) where T <: IntegerUnion
Vector(x::PermGroupElem, n::Int = x.parent.deg)
```

Return the list of length `n`

that contains `x(i)`

at position `i`

. If not specified, `T`

is set as `Int`

.

**Examples**

```
julia> pi = cperm(1:3)
(1,2,3)
julia> Vector(pi)
3-element Vector{Int64}:
2
3
1
julia> Vector(pi, 2)
2-element Vector{Int64}:
2
3
julia> Vector(pi, 4)
4-element Vector{Int64}:
2
3
1
4
julia> Vector{ZZRingElem}(pi, 2)
2-element Vector{ZZRingElem}:
2
3
```

## Operations on permutations

`sign`

— Method`sign(g::PermGroupElem) -> Int`

Return the sign of the permutation `g`

.

The sign of a permutation $g$ is defined as $(-1)^k$ where $k$ is the number of cycles of $g$ of even length.

**Examples**

```
julia> sign(cperm(1:2))
-1
julia> sign(cperm(1:3))
1
```

`isodd`

— Method`isodd(g::PermGroupElem)`

Return `true`

if the permutation `g`

is odd, `false`

otherwise.

A permutation is odd if it has an odd number of cycles of even length. Equivalently, a permutation is odd if it has sign $-1$.

**Examples**

```
julia> isodd(cperm(1:2))
true
julia> isodd(cperm(1:3))
false
julia> isodd(cperm(1:2,3:4))
false
```

`iseven`

— Method`iseven(g::PermGroupElem)`

Return `true`

if the permutation `g`

is even, `false`

otherwise.

A permutation is even if it has an even number of cycles of even length. Equivalently, a permutation is even if it has sign $+1$.

**Examples**

```
julia> iseven(cperm(1:2))
false
julia> iseven(cperm(1:3))
true
julia> iseven(cperm(1:2,3:4))
true
```

`cycle_structure`

— Method`cycle_structure(g::PermGroupElem) -> CycleType`

Return the cycle structure of the permutation `g`

as a cycle type. A cycle type behaves similar to a vector of pairs `k => n`

indicating that there are `n`

cycles of length `k`

.

**Examples**

```
julia> g = cperm(1:3, 4:5, 6:7, 8:10, 11:15)
(1,2,3)(4,5)(6,7)(8,9,10)(11,12,13,14,15)
julia> cycle_structure(g)
3-element Oscar.CycleType:
2 => 2
3 => 2
5 => 1
julia> g = cperm()
()
julia> cycle_structure(g)
1-element Oscar.CycleType:
1 => 1
```

## Permutations as functions

A permutation can be viewed as a function on the set `{1,...,n}`

, hence it can be evaluated on integers.

The multiplication between permutations works from the left to the right. So, if `x`

and `y`

are permutations and `n`

is an integer, then `(x*y)(n) = (y(x(n))`

, NOT `x(y(n))`

. This works also if the argument is not in the range `1:n`

; in such a case, the output coincides with the input.

```
julia> x = cperm([1,2,3,4,5]);
julia> x(2)
3
julia> x(6)
6
```

## Cycle structures

For a permutation, its cycle structure `cycle_structure`

determines the degree, order, number of moved points, sign.

`degree`

— Method`degree(c::CycleType) -> Int`

Return the degree of the permutations with cycle structure `c`

.

**Examples**

```
julia> g = symmetric_group(3);
julia> all(x -> degree(cycle_structure(x)) == degree(g), gens(g))
true
```

`iseven`

— Method`iseven(c::CycleType) -> Bool`

Return whether the permutations with cycle structure `c`

are even.

**Examples**

```
julia> g = symmetric_group(3);
julia> all(x -> iseven(cycle_structure(x)) == iseven(x), gens(g))
true
```

`isodd`

— Method`isodd(c::CycleType) -> Bool`

Return whether the permutations with cycle structure `c`

are odd.

**Examples**

```
julia> g = symmetric_group(3);
julia> all(x -> isodd(cycle_structure(x)) == isodd(x), gens(g))
true
```

`order`

— Method`order(::Type{T} = ZZRingElem, c::CycleType) where T <: IntegerUnion`

Return the order of the permutations with cycle structure `c`

.

**Examples**

```
julia> g = symmetric_group(3);
julia> all(x -> order(cycle_structure(x)) == order(x), gens(g))
true
```

`sign`

— Method`sign(c::CycleType) -> Int`

Return the sign of the permutations with cycle structure `c`

.

**Examples**

```
julia> g = symmetric_group(3);
julia> all(x -> sign(cycle_structure(x)) == sign(x), gens(g))
true
```

`cycle_structures`

— Method`cycle_structures(G::PermGroup) -> Set{CycleType}`

Return the set of cycle structures of elements in `G`

, see `cycle_structure`

.

**Examples**

```
julia> g = symmetric_group(3);
julia> sort!(collect(cycle_structures(g)))
3-element Vector{Oscar.CycleType}:
[1 => 1, 2 => 1]
[1 => 3]
[3 => 1]
```