Global

Members

globalRepo :PermutationRepository

Description:

  • Singleton instance of the PermutationRepository. All permutation operations should typically go through this global instance to ensure consistent ID management and memory optimization.

Source:

Singleton instance of the PermutationRepository. All permutation operations should typically go through this global instance to ensure consistent ID management and memory optimization.

Type:

Methods

analyzeGenerators(candidateIds) → {Object}

Description:

  • Analyzes a list of candidate generators to determine a minimal (fundamental) generating set. It uses the Schreier-Sims Algorithm to identify and separate redundant generators.

Source:
Parameters:
Name Type Description
candidateIds Array.<number> | PermutationSet

An array of permutation IDs that are potential generators.
Returns:

An object containing:

  • fundamental: An array of permutation IDs that form a minimal generating set.
  • redundant: An array of permutation IDs that are generated by the fundamental set.
  • ssa: The SchreierSimsAlgorithm instance computed from the fundamental generators.
Type
Object

areIsomorphic(groupA, groupB) → {number}

Description:

  • Heuristically checks if two groups groupA and groupB are isomorphic. This function compares structural invariants (order, abelian-ness, derived series length). It cannot definitively prove isomorphism without constructing an explicit isomorphism map, but it can reliably prove non-isomorphism and provide a strong indication for isomorphism.

Source:
Parameters:
Name Type Description
groupA PermutationSet | SchreierSimsAlgorithm

The first group.
groupB PermutationSet | SchreierSimsAlgorithm

The second group.
Returns:

1 (Isomorphic, if strictly proven - rare), 0 (Not isomorphic, strictly proven), -1 (Uncertain - heuristically likely isomorphic but not strictly proven).

Type
number

calcApproxOrder(perm, limitopt) → {number}

Description:

  • Computes an approximate order of a permutation by finding the LCM of its cycle lengths. If the order exceeds limit, it returns limit+1 as a sentinel value.

Source:
Parameters:
Name Type Attributes Default Description
perm *

The permutation array.
limit number <optional>
 60

The upper limit for the order.
Returns:
  • The approximate order of the permutation.
Type
number

createAlternating(n) → {PermutationSet}

Description:

  • Creates generators for the Alternating Group A_n. Contains even permutations. Order: n! / 2.

    Generators used: 3-cycles of the form (1 2 i) for i = 3..n [0-based: (0 1 i) for i=2..n-1].

Source:
Parameters:
Name Type Description
n number

The degree.
Returns:

A set containing n-2 generators.

Type
PermutationSet

createCyclic(n) → {PermutationSet}

Description:

  • Creates generators for the Cyclic Group C_n. Order: n.

    Generator used: One cycle (1 2 ... n) [0-based: (0 1 ... n-1)].

Source:
Parameters:
Name Type Description
n number

The degree.
Returns:

A set containing 1 generator.

Type
PermutationSet

createDihedral(n) → {PermutationSet}

Description:

  • Creates generators for the Dihedral Group D_n. Symmetries of a regular n-gon. Order: 2n.

    Generators used:

    1. Rotation r: (1 2 ... n)
    2. Reflection s: Fixes 1, maps k -> n-k+2 (mod n check) 0-based logic: Fixes 0, Maps k -> -k mod n.
Source:
Parameters:
Name Type Description
n number

The number of vertices.
Returns:

A set containing 2 generators.

Type
PermutationSet

createDirectProduct(groupA, groupB, …extraGroups) → {PermutationSet}

Description:

  • Creates the Direct Product of two groups: G x H. The resulting group acts on disjoint sets of points. Degree = Degree(G) + Degree(H).

Source:
Parameters:
Name Type Attributes Description
groupA PermutationSet

Generators for group G.
groupB PermutationSet

Generators for group H.
extraGroups PermutationSet <repeatable>

Additional groups to include in the direct product.
Returns:

Generators for G x H.

Type
PermutationSet

createFromCycleStrings(cyclesStrArr, degreeopt) → {PermutationSet}

Description:

  • Creates a generator set from a list of cycle strings. Convenient wrapper for parsing multiple permutations and registering them.

Source:
Parameters:
Name Type Attributes Default Description
cyclesStrArr Array.<string>

Array of strings, e.g. ["(1 2 3)", "(1 2)"].
degree number <optional>
 0

Force a specific degree. If 0, auto-detected per string (max).
Returns:

The set of generators.

Type
PermutationSet

createFromRawArrays(arrays) → {PermutationSet}

Description:

  • Creates a group from raw integer arrays. Registers each raw permutation array into the global repository and returns a PermutationSet of their IDs. Useful for loading from JSON or UI input.

Source:
Parameters:
Name Type Description
arrays Array.<(Int32Array|Array.<number>)>

An array of raw permutation arrays (e.g., [[0,1,2],[1,0,2]]).
Returns:

A PermutationSet containing the registered permutation IDs.

Type
PermutationSet

createIcosahedral() → {PermutationSet}

Description:

  • Icosahedral Group (Rotations of a regular icosahedron). Isomorphic to A5. Order 60.

Source:
Returns:

A PermutationSet representing the generators of the Icosahedral Group.

Type
PermutationSet

createKleinFour() → {PermutationSet}

Description:

  • Creates generators for the Klein Four-Group V_4. A subgroup of S_4 isomorphic to C_2 x C_2. Order: 4.

    Generators used:

    1. (1 2)(3 4) [0-based: (0 1)(2 3)]
    2. (1 3)(2 4) [0-based: (0 2)(1 3)]
Source:
Returns:

A set containing 2 generators on 4 points.

Type
PermutationSet

createOctahedral() → {PermutationSet}

Description:

  • Octahedral Group (Rotations of a regular octahedron). Isomorphic to S4. Order 24.

Source:
Returns:

A PermutationSet representing the generators of the Octahedral Group.

Type
PermutationSet

createQuaternion() → {PermutationSet}

Description:

  • Creates generators for the Quaternion Group Q8. Order: 8. Non-abelian. Defined via Regular Representation in S8. Elements: {1, i, j, k, -1, -i, -j, -k}

Source:
Returns:
Type
PermutationSet

createSymmetric(n) → {PermutationSet}

Description:

  • Creates generators for the Symmetric Group S_n. Contains all n! permutations of n elements.

    Generators used:

    1. Transposition (1 2) [0-based: (0 1)]
    2. Long Cycle (1 2 ... n) [0-based: (0 1 ... n-1)]
Source:
Parameters:
Name Type Description
n number

The degree (number of points).
Returns:

A set containing the generators.

Type
PermutationSet

createTetrahedral() → {PermutationSet}

Description:

  • Tetrahedral Group (Rotations of a regular tetrahedron). Isomorphic to A4. Order 12.

Source:
Returns:

A PermutationSet representing the generators of the Tetrahedral Group.

Type
PermutationSet

createTrivial() → {PermutationSet}

Description:

  • Creates the Trivial Group (Identity).

Source:
Returns:
Type
PermutationSet

decomposeToCycles(perm) → {string}

Description:

  • Decomposes a permutation into disjoint cycle notation string. Uses 1-based indexing for the output string.

Source:
Parameters:
Name Type Description
perm number | Int32Array | Array.<number>

Permutation ID (in globalRepo) or raw array.
Returns:

Cycle notation, e.g., "(1 2 3)(4 5)". Returns "()" for identity.

Type
string

findCoxeterLikeGenerators(inputGenerators, optionsopt) → {PermutationSet}

Description:

  • Attempts to find a set of generators that mimic a Coxeter system (strong generating set of involutions).

Source:
Parameters:
Name Type Attributes Description
inputGenerators PermutationSet | Array.<number>

The initial generators defining the group.
options Object <optional>

Configuration options.

Properties
Name Type Attributes Default Description
beamWidth number <optional>
 50

Number of candidates to keep in beam search.
generations number <optional>
 30

Number of mixing generations.
forcedBase Array.<number> <optional>

Force a specific base order.
Returns:

A new set of generators.

Type
PermutationSet

findLowOrderGenerators(inputGenerators, maxOrder, maxSearchSizeopt) → {PermutationSet|null}

Description:

  • Attempts to find a new set of generators for the group where every generator has an order less than or equal to maxOrder.

    It explores the group structure (via BFS) to find candidate elements of low order. If it finds enough low-order elements to generate the original group (verified by SSA), it returns this new set. Otherwise, returns null.

Source:
Parameters:
Name Type Attributes Default Description
inputGenerators PermutationSet | Array.<number>

The original generators of the group.
maxOrder number

The maximum allowed order for the new generators.
maxSearchSize number <optional>
 50000

Limit on the number of group elements to explore during search.
Returns:

A new PermutationSet if successful, or null if failed.

Type
PermutationSet | null

generateCayleyGraphForPlotly(inputIds, customConfigopt, extraGeneratorsopt) → {CayleyGraphData|Object}

Description:

  • Generates the graph data structure for a Cayley graph, including nodes, links, and a physics simulator. This function can return either a full data structure for advanced usage or a Plotly-ready frame.

Source:
See:
Parameters:
Name Type Attributes Default Description
inputIds Array.<number> | PermutationSet

Array of generator IDs used to construct the group.
customConfig Partial.<_CayleyGraphConfig> <optional>
 {}

Optional physics configuration overrides.
extraGenerators Array.<number> | PermutationSet <optional>
 []

Optional additional generators to visualize but exclude from physics forces.
Throws:

If no generators are provided.

Type
Error
Returns:

Returns a CayleyGraphData object if config.advancedMode is true, otherwise returns a Plotly-compatible object {data, layout, nameMap}.

Type
CayleyGraphData | Object

generateGroup(generators) → {PermutationSet}

Description:

  • Generates a subgroup from a set of generator permutations. This method uses an iterative closure approach by repeatedly multiplying the current group by the generators until no new elements are found.

Source:
Parameters:
Name Type Description
generators PermutationSet | Array.<number> | SchreierSimsAlgorithm

A PermutationSet or an array of permutation IDs or a SchreierSimsAlgorithm instance to generate the group from.
Throws:

If generators is an unknown type.

Type
Error
Returns:

The fully generated subgroup (isGroup=true).

Type
PermutationSet

generateMultiplicationTable(inputIds, nameMapopt) → {Object}

Description:

  • Generates a Multiplication (Cayley) Table for a group. inputIds are treated as candidate generators. The function will determine a fundamental set of generators, expand the group to all its elements, and generate names for them. The table will represent the full group.

    return an object A 2D array where matrix[row][col] is the permutation ID of rowElement * colElement. A 2D array where grid[row][col] is the algebraic name (string) of rowElement * colElement. A Map where keys are permutation IDs and values are their 1-based cycle notation strings (e.g., "(1 2 3)"). An HTML string representation of the Cayley table with semantic coloring and tooltips.

Source:
See:
Parameters:
Name Type Attributes Default Description
inputIds Array.<number>

An array of candidate generator IDs.
nameMap Map.<number, string> <optional>
 null

Optional. A custom map of all permutation IDs to their display names. Use generateNames to generate.
Throws:

If nameMap is provided in manual mode but is incomplete (missing names for inputIds).

Type
Error
Returns:

An object containing the generated table data.

Type
Object

generateNames(allElementIds, generatorIds, genLabelsopt) → {Map.<number, string>}

Description:

  • Generates human-readable algebraic names for all group elements (e.g., 'e', 'a', 'b', 'ab', 'a^2'). This function uses a Breadth-First Search (BFS) approach, starting from the identity and generators, to construct the shortest and most intuitive names based on generator products.

Source:
Parameters:
Name Type Attributes Description
allElementIds Array.<number> | PermutationSet

A sorted list of all unique permutation IDs belonging to the group.
generatorIds Array.<number> | PermutationSet

A list of permutation IDs that are the fundamental generators of the group.
genLabels Array.<string> <optional>

A list of strings that are the labels for the generators. Default to undefined means to use a,b,c,...
Returns:

A Map where keys are permutation IDs and values are their corresponding generated algebraic names.

Type
Map.<number, string>

getCommutatorSubgroup(group) → {SchreierSimsAlgorithm}

Description:

  • Computes the commutator subgroup G' = [G, G] of a group G. This subgroup is generated by all commutators [g1, g2] = g1^-1 * g2^-1 * g1 * g2 for g1, g2 in G.

Source:
Parameters:
Name Type Description
group PermutationSet | SchreierSimsAlgorithm

The group G.
Returns:

The SchreierSimsAlgorithm instance representing the commutator subgroup.

Type
SchreierSimsAlgorithm

getLowerCentralSeries(group) → {Array.<SchreierSimsAlgorithm>}

Description:

  • Computes the lower central series of a group G. The series is defined recursively as: G_0 = G G_{i+1} = [G_i, G] (the mixed commutator subgroup of G_i and G). The series terminates when G_{i+1} = G_i or G_i = {e}.

Source:
Parameters:
Name Type Description
group PermutationSet | SchreierSimsAlgorithm

The group G.
Returns:

An array of SchreierSimsAlgorithm instances, representing the subgroups in the lower central series: [G_0, G_1, ..., G_k].

Type
Array.<SchreierSimsAlgorithm>

getMixedCommutatorSubgroup(groupG, subA, subB) → {SchreierSimsAlgorithm}

Description:

  • Computes the mixed commutator subgroup [subA, subB] of two subgroups subA and subB within a larger group groupG. The result is the normal closure of all commutators [a, b] (where a is from subA and b is from subB) within the group groupG.

Source:
Parameters:
Name Type Description
groupG PermutationSet | SchreierSimsAlgorithm

The containing parent group G, used for computing the normal closure.
subA PermutationSet | SchreierSimsAlgorithm

The first subgroup A.
subB PermutationSet | SchreierSimsAlgorithm

The second subgroup B.
Returns:

The SchreierSimsAlgorithm instance representing the mixed commutator subgroup [A, B].

Type
SchreierSimsAlgorithm

getNormalClosure(groupG, subsetS) → {SchreierSimsAlgorithm}

Description:

  • Computes the normal closure of a subset subsetS within the group groupG. The normal closure is the smallest normal subgroup of groupG that contains subsetS. It is generated by all conjugates of elements of subsetS by elements of groupG.

Source:
Parameters:
Name Type Description
groupG PermutationSet | SchreierSimsAlgorithm

The containing group G.
subsetS PermutationSet | Array.<number> | number

The subset S (generators, array of IDs, or single ID).
Returns:

The SchreierSimsAlgorithm instance representing the normal closure.

Type
SchreierSimsAlgorithm

getQuotientStructure(groupG, normalN, maxIndexopt) → {QuotientGroupMap}

Description:

  • Computes the structure of the quotient group G/N, along with a mapping that lifts elements from G/N back to representatives in G. This function is computationally intensive and only feasible for quotient groups with a small index [G:N].

Source:
Parameters:
Name Type Attributes Default Description
groupG PermutationSet | SchreierSimsAlgorithm

The group G.
normalN PermutationSet | SchreierSimsAlgorithm

The normal subgroup N of G.
maxIndex number <optional>
 2000

The maximum allowed index [G:N] for explicit construction.
Throws:

If N is not a normal subgroup of G, or if [G:N] exceeds maxIndex.

Type
Error
Returns:

An object containing the quotient group (as PermutationSet) and an array of representatives for each coset.

Type
QuotientGroupMap

getSylowSubgroup(group, p) → {PermutationSet}

Description:

  • Computes a Sylow p-subgroup of a given group G. A Sylow p-subgroup is a maximal p-subgroup of G, with order p^k where p^k divides |G| and p^(k+1) does not. The algorithm uses a randomized greedy approach with restarts.

Source:
Parameters:
Name Type Description
group PermutationSet | SchreierSimsAlgorithm

The group G for which to find a Sylow p-subgroup.
p number

The prime number p.
Throws:

If the algorithm fails to construct a Sylow p-subgroup within the configured random search limits.

Type
Error
Returns:

A PermutationSet containing the generators of a Sylow p-subgroup.

Type
PermutationSet

isNilpotent(group) → {number}

Description:

  • Checks if a group is nilpotent. A group G is nilpotent if its lower central series terminates at the trivial group {e}. Every nilpotent group is solvable.

Source:
Parameters:
Name Type Description
group PermutationSet | SchreierSimsAlgorithm

The group G to check for nilpotency.
Returns:

1 (Nilpotent), 0 (Not Nilpotent).

Type
number

isNormal(superGroup, normalN) → {boolean}

Description:

  • Checks if normalN is a normal subgroup of superGroup (N ◁ G). This is verified by checking if for every generator g of superGroup and every generator n of normalN, the conjugate g * n * g^-1 is an element of normalN.

Source:
Parameters:
Name Type Description
superGroup PermutationSet | SchreierSimsAlgorithm

The supergroup G.
normalN PermutationSet | SchreierSimsAlgorithm

The potential normal subgroup N.
Returns:

True if N is a normal subgroup of G, false otherwise.

Type
boolean

isSimple(group, randomTestsopt) → {number}

Description:

  • Checks if a group is simple. A group G is simple if its only normal subgroups are the trivial group {e} and G itself. This function uses a probabilistic approach for non-abelian groups and may return "uncertain" for large groups.

Source:
Parameters:
Name Type Attributes Default Description
group PermutationSet | SchreierSimsAlgorithm

The group G to check for simplicity.
randomTests number <optional>
 10

Number of random conjugates to test for non-abelian groups. Higher values increase confidence but also computation time.
Returns:

1 (Proven Simple), 0 (Proven Not Simple), -1 (Uncertain - heuristically likely simple but not strictly proven).

Type
number

isSolvable(group) → {boolean}

Description:

  • Checks if a group is solvable. A group G is solvable if its derived series terminates in the trivial group {e}. The derived series is G^(0) = G, G^(n+1) = [G^(n), G^(n)].

Source:
Parameters:
Name Type Description
group PermutationSet | SchreierSimsAlgorithm

The group G to check for solvability.
Returns:

True if the group is solvable, false otherwise.

Type
boolean

isSubgroup(superGroup, subGroup) → {boolean}

Description:

  • Checks if subGroup is a subgroup of superGroup. This is determined by verifying that all generators of subGroup are contained within superGroup.

Source:
Parameters:
Name Type Description
superGroup PermutationSet | SchreierSimsAlgorithm

The potential supergroup G.
subGroup PermutationSet | SchreierSimsAlgorithm

The potential subgroup H.
Returns:

True if H is a subgroup of G, false otherwise.

Type
boolean

parseCycles(str, degreeopt) → {Int32Array}

Description:

  • Parses a string in cycle notation into a flat permutation array. Supports standard disjoint cycle notation, e.g., "(1 2 3)(4 5)".

    Assumptions:

    • Input uses 1-based indexing (standard mathematical notation).
    • Output is 0-based Int32Array.
Source:
Parameters:
Name Type Attributes Default Description
str string

The cycle string (e.g., "(1 2 3)").
degree number <optional>
 0

The required degree (size) of the permutation. If 0, inferred from max element.
Returns:

The permutation in array form [p[0], p[1], ...].

Type
Int32Array

resetGlobalRepo()

Description:

  • Resets the global permutation repository. This function clears all registered permutations and re-initializes globalRepo to a new empty PermutationRepository instance. Use with caution, as all previously obtained permutation IDs will become invalid.

Source:

Type Definitions

CayleyGraphData

Source:
Properties:
Name Type Description
nodes Array.<object>

Array of node objects with id, name, x, y, z, vx, vy, vz properties.
links Array.<object>

Array of link objects with source, target, genId, color, order, isDirected properties.
legend Array.<object>

Array of legend objects with label, color, genId properties.
simulator VisualizerCayleyForceSimulator

The force simulator instance.
config _CayleyGraphConfig

The effective physics configuration used.
nameMap Map.<number, string>

the used nameMap
Type:
  • object

_CayleyGraphConfig

Source:
See:
Properties:
Name Type Description
d0 number

Base distance factor. Used to determine ideal edge length (d0 / order).
repulsion number

Strength of the Coulomb-like repulsive force between all nodes.
edgeStrength number

Spring constant for edges directly connecting elements (Hooke's law).
chordStrength number

Spring constant for 'chord' edges in cycles, maintaining their geometric shape.
planarStrength number

Strength of the force that flattens cycles onto a plane.
convexityStrength number

Strength of the force that pushes nodes away from the center of a cycle, maintaining convexity.
initialOffsetDist number

Magnitude of random initial displacement applied to cycle groups to untangle them during setup.
decay number

Velocity damping factor, reducing oscillation.
centerPull number

Strength of the gravitational force pulling all nodes towards the origin.
timeStep number

Simulation time step for integration.
jitterMax number

Maximum random displacement applied to nodes during the initial annealing phase.
dynamicAngleUpdateRate number

How often (in ticks) to recalculate average angles for chord length adjustment. Set to 0 to disable.
warmupRuns number

Number of simulation ticks to run during the warmup phase for initial layout.
advancedMode boolean

If true, returns detailed simulator objects; otherwise, returns only Plotly frame.
nameMap Map.<number, string>

the nameMap to use for the full group elements. Use generateNames to generate.
rewriteToStrongGenerators boolean

use SSA to rewrite generators to StrongGenerators.
Type:
  • object