PermutationGroup

Documentation of 'cc.redberry.core.groups.permutations.PermutationGroup' Java class PermutationGroup

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java.lang.Object
- cc.redberry.core.groups.permutations.PermutationGroup

All Implemented Interfaces:

java.lang.Iterable<Permutation>
```
public final class PermutationGroup
extends java.lang.Object
implements java.lang.Iterable<Permutation>
```
Implementation of permutation group; this class provides a number of methods for work wih permutation groups, including membership testing, coset enumeration, searching for centralizers, stabilizers, etc (for details see method summary). The instances of this class are immutable. The iterator returned by this class's iterator() method iterates over all elements of this group.
Example

The following example gives a brief overview of the basic usage of PermutationGroup
```
//Construct permutation group of degree 13 with two generators (written in one-line notation)
 PermutationGroup pg =  PermutationGroup.createPermutationGroup(
     Permutations.createPermutation(9, 1, 2, 0, 4, 8, 5, 11, 6, 3, 10, 12, 7),
     Permutations.createPermutation(2, 0, 1, 8, 3, 5, 7, 11, 4, 12, 9, 6, 10));
 //this group is transitive
 assert pg.isTransitive();
 //its order = 5616
 System.out.println(pg.order());
 //Create alternating group Alt(13)
 PermutationGroup alt13 = PermutationGroup.alternatingGroup(13);
 //its order = 3113510400
 System.out.println(alt13.order());
 assert alt13.containsSubgroup(pg);
 //Direct product of two groups
 PermutationGroup pp = pg.directProduct(PermutationGroup.symmetricGroup(8));
 //Setwise stabilizer
 PermutationGroup sw = pp.setwiseStabilizer(1, 2, 3, 9, 10, 11, 12, 3, 14, 15, 16, 17, 18);
 assert pp.containsSubgroup(sw);
 //its order = 17280
 System.out.println(sw.order());
 //Center of this stabilizer
 PermutationGroup center = sw.center();
 //it is abelian group
 assert center.isAbelian();
 //generators of center
 System.out.println(center.generators());
 //[+{}, +{{19, 20}}, +{{2, 10}, {3, 9}, {6, 8}, {11, 12}}]
 //orbits of center
 int[][] orbits = center.orbits();
 for (int[] orbit : orbits)
 if (orbit.length != 1)
 System.out.print(Arrays.toString(orbit));
 //[2, 10], [3, 9], [6, 8], [11, 12], [19, 20]
 
```
Implementation and complexity

The implementation is based on base and strong generating set (BSGS), which is constructed using Schreier-Sims algorithm (AlgorithmsBase.SchreierSimsAlgorithm(java.util.ArrayList)). Schreier-Sims algorithm has O(n^6 +k*n^2) complexity (where n is a degree of group), which can be crucial for groups with large bases. Since not all methods require BSGS, the BSGS structure of PermutationGroup is lazy initialized, i.e. its initialization occurs on the first invocation of method that uses BSGS.

Structural calculations. Generally, all structural calculations have polynomial time complexity. The polynomial-time operations include: calculation of orbits (do not require BSGS); membership testing; calculation of order, base and strong generating set, pointwise stabilizers, union and direct product of groups; tests for commutativity, transitivity, Alt(n) and Sym(n) testing; calculation of normal closure and derived subgroup.

Backtrack search. On the other hand, the algorithms that use backtrack search methods have exponential complexity in the worst case, which also hardly depends on the input. This algorithms include: calculation of setwise stabilizers, intersections of groups, coset representatives, centralizers. The exception is the calculation of center of group which is always polynomial.
Since:

1.1.6

See Also:

Permutation, AlgorithmsBase, AlgorithmsBacktrack, BacktrackSearch

Field Summary

Fields
Modifier and Type Field and Description

static PermutationGroup TRIVIAL_GROUP

Fields
Modifier and Type	Field and Description
`static PermutationGroup`	`TRIVIAL_GROUP`

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`static PermutationGroup`	`alternatingGroup(int degree)` Creates alternating group of specified degree.
`static PermutationGroup`	`antisymmetricGroup(int degree)` Creates symmetric group of specified degree, where all odd permutations are antisymmetries.
`PermutationGroup`	`center()` Computes center of this group.
`PermutationGroup`	`centralizerOf(Permutation permutation)` Computes centralizer of specified permutation.
`PermutationGroup`	`centralizerOf(PermutationGroup subgroup)` Computes centralizer of specified subgroup.
`PermutationGroup`	`commutator(PermutationGroup group)` Returns a commutator of this group with specified group.
`PermutationGroup`	`conjugate(Permutation permutation)` Returns the conjugate permutation group of `this` with the specified permutation (this ^ permutation).
`boolean`	`containsSubgroup(PermutationGroup subgroup)` Returns true if specified group is a subgroup of this.
`static PermutationGroup`	`createPermutationGroup(java.util.List<Permutation> generators)` Creates permutation group with a given generating set.
`static PermutationGroup`	`createPermutationGroup(Permutation... generators)` Creates permutation group with a given generating set.
`static PermutationGroup`	`createPermutationGroupFromBSGS(java.util.List<BSGSElement> bsgs)` Creates permutation group with a given base and strong generating set.
`int`	`degree()` Returns the natural degree of this group, i.e.
`PermutationGroup`	`derivedSubgroup()` Returns a derived subgroup, i.e.
`PermutationGroup`	`directProduct(PermutationGroup group)` Returns direct product of this group and specified group.
`boolean`	`equals(java.lang.Object obj)` Returns true if specified group is equals to this group, i.e.
`java.util.List<Permutation>`	`generators()` Returns an unmodifiable list of group generators.
`int[]`	`getBase()` Returns base of this group.
`java.util.List<BSGSElement>`	`getBSGS()` Returns base and strong generating set of this group.
`java.util.ArrayList<BSGSCandidateElement>`	`getBSGSCandidate()` Returns a mutable copy of base and strong generating set.
`int[]`	`getPositionsInOrbits()` Returns positions of points in array of orbits, i.e.
`int`	`indexOfOrbit(int point)` Returns an index of orbit of this point in the array of all orbits returned by method `orbits()`, i.e.
`PermutationGroup`	`intersection(PermutationGroup oth)` Returns an intersection of this group with specified group.
`boolean`	`isAbelian()` Returns true if this group is abelian and false otherwise.
`boolean`	`isAlternating()` Returns true is this group is natural alternating group Alt(degree) and false otherwise.
`boolean`	`isRegular()` Returns true if this group is regular (transitive and its order equals to degree) and false otherwise,
`boolean`	`isSymmetric()` Returns true if this group is natural symmetric group and false otherwise.
`boolean`	`isTransitive()` Returns true if this group is transitive under the action on the set of its moved points and false otherwise.
`boolean`	`isTransitive(int from, int to)` Returns true if this group acts transitively on the array `[from, from + 1,...,to-1]` and false if not.
`boolean`	`isTrivial()` Returns true if this group is trivial and false otherwise.
`java.util.Iterator<Permutation>`	`iterator()` Returns an iterator over all elements in this group.
`Permutation[]`	`leftCosetRepresentatives(PermutationGroup subgroup)` Returns a set of left coset representatives of a given subgroup in this group (by definition, left coset of subgroup K have a form g*K); each representative is minimal in its coset under the ordering returned by `this.ordering()`.
`Permutation`	`leftTransversalOf(PermutationGroup subgroup, Permutation element)` Returns a unique left coset representative of specified element; the returned representative will be minimal in its coset under the ordering returned by `ordering()`.
`BacktrackSearch`	`mapping(int[] from, int[] to)` Returns an output port of permutations that preserves specified mapping between points.
`Permutation`	`mapping(int from, int to)` Returns some permutation that maps point from onto point to or `null` if no such permutation exists.
`boolean`	`membershipTest(java.util.Collection<Permutation> permutations)` Returns true if all specified permutations are members of this group and false otherwise.
`boolean`	`membershipTest(Permutation permutation)` Returns true if specified permutation is member of this group and false otherwise.
`PermutationGroup`	`normalClosureOf(PermutationGroup subgroup)` Calculates normal closure of specified subgroup.
`int[]`	`orbit(int... points)` Returns the orbit of specified set of points.
`int[]`	`orbit(int point)` Returns the orbit of specified point.
`int[][]`	`orbits()` Returns an array of all orbits.
`int`	`orbitSize(int point)` Returns size of orbit of specified point.
`java.math.BigInteger`	`order()` Returns the order of this group, i.e.
`InducedOrdering`	`ordering()` Returns an ordering on Ω(degree) induced by a base of this group.
`PermutationGroup`	`pointwiseStabilizer(int... set)` Calculates a pointwise stabilizer of specified set of points.
`PermutationGroup`	`pointwiseStabilizerRestricted(int... set)` Calculates a group which isomorphic to a pointwise stabilizer of specified set but acts on points that are not stabilized, i.e.
`Permutation`	`randomElement()` Returns uniformly distributed random permutation from this group.
`Permutation`	`randomElement(org.apache.commons.math3.random.RandomGenerator generator)` Returns uniformly distributed random permutation from this group.
`java.util.List<Permutation>`	`randomSource()` Returns a random source of permutations in this group.
`Permutation[]`	`rightCosetRepresentatives(PermutationGroup subgroup)` Returns a set of right coset representatives of a given subgroup in this group (by definition, right coset of subgroup K have a form K*g).
`PermutationGroup`	`setwiseStabilizer(int... set)` Calculates a setwise stabilizer of specified set of points.
`static PermutationGroup`	`symmetricGroup(int degree)` Creates symmetric group of specified degree.
`java.lang.String`	`toString()`
`java.lang.String`	`toStringJava()` Returns string to directly paste into Java code.
`static PermutationGroup`	`trivialGroup()`
`PermutationGroup`	`union(java.util.List<Permutation> generators)` Extends this group with specified generators, i.e.
`PermutationGroup`	`union(Permutation... generators)` Extends this group with specified generators, i.e.
`PermutationGroup`	`union(PermutationGroup group)` Returns a union of this and specified group, i.e.

Methods inherited from class java.lang.Object
getClass, hashCode, notify, notifyAll, wait, wait, wait

Methods inherited from interface java.lang.Iterable
forEach, spliterator

- Field Detail
  - TRIVIAL_GROUP
```
public static final PermutationGroup TRIVIAL_GROUP
```
- Method Detail
  - trivialGroup
```
public static PermutationGroup trivialGroup()
```
  - createPermutationGroup
```
public static PermutationGroup createPermutationGroup(Permutation... generators)
```
    Creates permutation group with a given generating set.
    
    Parameters:
    
    generators - generating set
  - createPermutationGroup
```
public static PermutationGroup createPermutationGroup(java.util.List<Permutation> generators)
```
    Creates permutation group with a given generating set.
    
    Parameters:
    
    generators - generating set
  - createPermutationGroupFromBSGS
```
public static PermutationGroup createPermutationGroupFromBSGS(java.util.List<BSGSElement> bsgs)
```
    Creates permutation group with a given base and strong generating set.
    
    Parameters:
    
    bsgs - base and strong generating set
  - symmetricGroup
```
public static PermutationGroup symmetricGroup(int degree)
```
    Creates symmetric group of specified degree. BSGS structure of symmetric group will be constructed in O(n^2) time.
    
    Parameters:
    
    degree - degree
    
    Returns:
    
    symmetric group of specified degree
    
    See Also:
    
    AlgorithmsBase.createSymmetricGroupBSGS(int)
  - antisymmetricGroup
```
public static PermutationGroup antisymmetricGroup(int degree)
```
    Creates symmetric group of specified degree, where all odd permutations are antisymmetries. BSGS structure of symmetric group will be constructed in O(n^2) time.
    
    Parameters:
    
    degree - degree
    
    Returns:
    
    antisymmetric group of specified degree
    
    See Also:
    
    AlgorithmsBase.createSymmetricGroupBSGS(int)
  - alternatingGroup
```
public static PermutationGroup alternatingGroup(int degree)
```
    Creates alternating group of specified degree. BSGS structure of alternating group will be constructed in O(n^2) time.
    
    Parameters:
    
    degree - degree
    
    Returns:
    
    alternating group of specified degree
    
    See Also:
    
    AlgorithmsBase.createAlternatingGroupBSGS(int)
  - getPositionsInOrbits
```
public int[] getPositionsInOrbits()
```
    Returns positions of points in array of orbits, i.e. for each point orbits()[getPositionsInOrbits()[point]] - is its orbit.
    
    Returns:
    
    positions of points in array of orbits
  - generators
```
public java.util.List<Permutation> generators()
```
    Returns an unmodifiable list of group generators.
    
    Returns:
    
    unmodifiable list of group generators
  - degree
```
public int degree()
```
    Returns the natural degree of this group, i.e. the largest point moved by this group plus one.
    
    Returns:
    
    the largest point moved by this group plus one
  - orbit
```
public int[] orbit(int point)
```
    Returns the orbit of specified point.
    
    Parameters:
    
    point - point
    
    Returns:
    
    orbit of specified point
  - orbitSize
```
public int orbitSize(int point)
```
    Returns size of orbit of specified point.
    
    Parameters:
    
    point - point
    
    Returns:
    
    size of orbit of specified point
  - orbit
```
public int[] orbit(int... points)
```
    Returns the orbit of specified set of points.
    
    Parameters:
    
    points - set of points
    
    Returns:
    
    orbit of specified set of points
  - orbits
```
public int[][] orbits()
```
    Returns an array of all orbits.
    
    Returns:
    
    an array of all orbits
  - indexOfOrbit
```
public int indexOfOrbit(int point)
```
    Returns an index of orbit of this point in the array of all orbits returned by method orbits(), i.e. orbits()[indexOfOrbit(point)] is orbit of specified point, or -1 if point >= degree().
    
    Parameters:
    
    point - point
    
    Returns:
    
    index of orbit of this point in the array of all orbits or -1 if point >= degree()
  - isTransitive
```
public boolean isTransitive()
```
    Returns true if this group is transitive under the action on the set of its moved points and false otherwise.
    
    Returns:
    
    true if this group is transitive under the action on the set of its moved points and false otherwise
  - isTransitive
```
public boolean isTransitive(int from,
                            int to)
```
    Returns true if this group acts transitively on the array [from, from + 1,...,to-1] and false if not.
    
    Returns:
    
    true if this group acts transitively on the array [from, from + 1,...,to-1] and false if not
  - isTrivial
```
public boolean isTrivial()
```
    Returns true if this group is trivial and false otherwise.
    
    Returns:
    
    true if this group is trivial and false otherwise
  - isAbelian
```
public boolean isAbelian()
```
    Returns true if this group is abelian and false otherwise.
    
    Returns:
    
    true if this group is abelian and false otherwise.
  - randomSource
```
public java.util.List<Permutation> randomSource()
```
    Returns a random source of permutations in this group.
    
    Returns:
    
    a random source of permutations in this group
    
    See Also:
    
    RandomPermutation.random(java.util.List, org.apache.commons.math3.random.RandomGenerator)
  - union
```
public PermutationGroup union(Permutation... generators)
```
    Extends this group with specified generators, i.e. returns a union of this group and a group generated by specified generators.
    
    Parameters:
    
    generators - new generators
    
    Returns:
    
    a group generated by this and specified generators
  - union
```
public PermutationGroup union(java.util.List<Permutation> generators)
```
    Extends this group with specified generators, i.e. returns a union of this group and a group generated by specified generators.
    
    Parameters:
    
    generators - new generators
    
    Returns:
    
    a group generated by this and specified generators
  - getBSGS
```
public java.util.List<BSGSElement> getBSGS()
```
    Returns base and strong generating set of this group.
    
    Returns:
    
    base and strong generating set of this group
  - getBase
```
public int[] getBase()
```
    Returns base of this group.
    
    Returns:
    
    base of this group
  - order
```
public java.math.BigInteger order()
```
    Returns the order of this group, i.e. the number of permutations in this group.
    
    Returns:
    
    the order of this group
  - ordering
```
public InducedOrdering ordering()
```
    Returns an ordering on Ω(degree) induced by a base of this group.
    
    Returns:
    
    ordering on Ω(degree) induced by a base of this group
  - membershipTest
```
public boolean membershipTest(Permutation permutation)
```
    Returns true if specified permutation is member of this group and false otherwise.
    
    Parameters:
    
    permutation - permutation
    
    Returns:
    
    true if specified permutation is member of this group and false otherwise
  - membershipTest
```
public boolean membershipTest(java.util.Collection<Permutation> permutations)
```
    Returns true if all specified permutations are members of this group and false otherwise.
    
    Parameters:
    
    permutations - permutations
    
    Returns:
    
    true if all specified permutations are members of this group and false otherwise
    
    See Also:
    
    membershipTest(Permutation)
  - randomElement
```
public Permutation randomElement()
```
    Returns uniformly distributed random permutation from this group.
    
    Returns:
    
    uniformly distributed random permutation from this group
  - randomElement
```
public Permutation randomElement(org.apache.commons.math3.random.RandomGenerator generator)
```
    Returns uniformly distributed random permutation from this group.
    
    Parameters:
    
    generator - random generator to be used in generation of random permutation
    
    Returns:
    
    uniformly distributed random permutation from this group
  - getBSGSCandidate
```
public java.util.ArrayList<BSGSCandidateElement> getBSGSCandidate()
```
    Returns a mutable copy of base and strong generating set.
    
    Returns:
    
    a mutable copy of base and strong generating set
  - isSymmetric
```
public boolean isSymmetric()
```
    Returns true if this group is natural symmetric group and false otherwise.
    
    Returns:
    
    true is this group is natural symmetric group and false otherwise
  - isAlternating
```
public boolean isAlternating()
```
    Returns true is this group is natural alternating group Alt(degree) and false otherwise.
    
    Returns:
    
    true is this group is natural alternating group Alt(degree) and false otherwise
  - isRegular
```
public boolean isRegular()
```
    Returns true if this group is regular (transitive and its order equals to degree) and false otherwise,
    
    Returns:
    
    true if this group is regular and false otherwise
  - pointwiseStabilizer
```
public PermutationGroup pointwiseStabilizer(int... set)
```
    Calculates a pointwise stabilizer of specified set of points.
    
    Parameters:
    
    set - set of points
    
    Returns:
    
    pointwise stabilizer of specified set of points.
  - pointwiseStabilizerRestricted
```
public PermutationGroup pointwiseStabilizerRestricted(int... set)
```
    Calculates a group which isomorphic to a pointwise stabilizer of specified set but acts on points that are not stabilized, i.e. the degree of the resulting group equal to this.degree() - set.length (under the assumption that set contains distinct points).
    
    Parameters:
    
    set - set of points
    
    Returns:
    
    pointwise stabilizer of specified set of points that acts on points which not contained in specified set
  - normalClosureOf
```
public PermutationGroup normalClosureOf(PermutationGroup subgroup)
```
    Calculates normal closure of specified subgroup. The algorithm follows NORMALCLOSURE (randomized version) described in Sec. 3.3.2 in [Holt05].
    
    Parameters:
    
    subgroup - subgroup of this
    
    Returns:
    
    normal closure
  - commutator
```
public PermutationGroup commutator(PermutationGroup group)
```
    Returns a commutator of this group with specified group.
    
    Parameters:
    
    group - permutation group
    
    Returns:
    
    commutator of this and specified group
  - derivedSubgroup
```
public PermutationGroup derivedSubgroup()
```
    Returns a derived subgroup, i.e. commutator subgroup of this with itself.
    
    Returns:
    
    derived subgroup, i.e. commutator subgroup of this with itself
  - setwiseStabilizer
```
public PermutationGroup setwiseStabilizer(int... set)
```
    Calculates a setwise stabilizer of specified set of points.
    
    Parameters:
    
    set - set of points
    
    Returns:
    
    setwise stabilizer of specified set of points.
  - containsSubgroup
```
public boolean containsSubgroup(PermutationGroup subgroup)
```
    Returns true if specified group is a subgroup of this.
    
    Parameters:
    
    subgroup - permutation group
    
    Returns:
    
    true if specified group is a subgroup of this
  - leftCosetRepresentatives
```
public Permutation[] leftCosetRepresentatives(PermutationGroup subgroup)
```
    Returns a set of left coset representatives of a given subgroup in this group (by definition, left coset of subgroup K have a form g*K); each representative is minimal in its coset under the ordering returned by this.ordering(). The number of these representatives is equals to this.order().divide(subgroup.order()).
    
    Parameters:
    
    subgroup - a subgroup of this group
    
    Returns:
    
    set of left coset representatives
    
    See Also:
    
    AlgorithmsBacktrack.leftCosetRepresentatives(java.util.List, java.util.List)
  - rightCosetRepresentatives
```
public Permutation[] rightCosetRepresentatives(PermutationGroup subgroup)
```
    Returns a set of right coset representatives of a given subgroup in this group (by definition, right coset of subgroup K have a form K*g). The number of these representatives is equals to this.order().divide(subgroup.order()). This method calculates left coset representatives using leftCosetRepresentatives(PermutationGroup) and inverse each representative. In contrast to leftCosetRepresentatives(PermutationGroup) each right coset representative is not necessary minimal in its coset.
    
    Parameters:
    
    subgroup - a subgroup of this group
    
    Returns:
    
    set of right coset representatives
    
    See Also:
    
    leftCosetRepresentatives(PermutationGroup)
  - leftTransversalOf
```
public Permutation leftTransversalOf(PermutationGroup subgroup,
                                     Permutation element)
```
    Returns a unique left coset representative of specified element; the returned representative will be minimal in its coset under the ordering returned by ordering().
    
    Parameters:
    
    subgroup - a subgroup of this group
    
    element - some element of this group
    
    See Also:
    
    AlgorithmsBacktrack.leftTransversalOf(Permutation, java.util.List, java.util.List)
  - union
```
public PermutationGroup union(PermutationGroup group)
```
    Returns a union of this and specified group, i.e. group which is generated by union of generators of this and specified group.
    
    Parameters:
    
    group - permutation group
    
    Returns:
    
    union of this and specified group
  - intersection
```
public PermutationGroup intersection(PermutationGroup oth)
```
    Returns an intersection of this group with specified group.
    
    Parameters:
    
    oth - permutation group
    
    Returns:
    
    intersections of groups
  - directProduct
```
public PermutationGroup directProduct(PermutationGroup group)
```
    Returns direct product of this group and specified group. This product is organized as follows: the initial segment of each permutation is equal to permutation taken from this, while the rest is taken from specified group.
    
    Parameters:
    
    group - another group
    
    Returns:
    
    direct product this × other
  - mapping
```
public Permutation mapping(int from,
                           int to)
```
    Returns some permutation that maps point from onto point to or null if no such permutation exists.
    
    Parameters:
    
    from - from point
    
    to - to point
    
    Returns:
    
    some permutation that maps point from onto point to or null if no such permutation exists
  - mapping
```
public BacktrackSearch mapping(int[] from,
                               int[] to)
```
    Returns an output port of permutations that preserves specified mapping between points. To be precise: for each permutation p returned by BacktrackSearch.take() and for all i ∈ {0..from.length}, it is guaranteed to be p.newIndexOf(from[i]) == to[i].
    Example:
    The following code
```
Permutation perm1 = Permutations.createPermutation(8, new int[][]{{1, 2, 3}});
 Permutation perm2 = Permutations.createPermutation(8, new int[][]{{3, 4, 5, 6, 7}});
 PermutationGroup pg = PermutationGroup.createPermutationGroup(perm1, perm2);
 BacktrackSearch mappings = pg.mapping(new int[]{7, 2, 1, 3}, new int[]{5, 3, 6, 1});
 Permutation perm;
 while ((perm = mappings.take()) != null)
 System.out.println(perm);
 
```
    will produce 3 permutations (in cycles notation):
        +{{1, 6, 2, 3}, {5, 7}}
        +{{1, 6, 7, 5, 4, 2, 3}}
        +{{1, 6, 4, 7, 5, 2, 3}}
    Parameters:
    
    from - points from
    
    to - points to
    
    Returns:
    
    output port of permutations that preserves specified mapping between points
    
    Throws:
    
    java.lang.IllegalArgumentException - if from.length != to.length
  - iterator
```
public java.util.Iterator<Permutation> iterator()
```
    Returns an iterator over all elements in this group.
    
    Specified by:
    
    iterator in interface java.lang.Iterable<Permutation>
    
    Returns:
    
    iterator over all elements in this group
  - centralizerOf
```
public PermutationGroup centralizerOf(Permutation permutation)
```
    Computes centralizer of specified permutation.
    
    Parameters:
    
    permutation - permutation
    
    Returns:
    
    centralizer of specified element
  - centralizerOf
```
public PermutationGroup centralizerOf(PermutationGroup subgroup)
```
    Computes centralizer of specified subgroup.
    
    Parameters:
    
    subgroup - a subgroup of this
    
    Returns:
    
    centralizer of specified subgroup
  - center
```
public PermutationGroup center()
```
    Computes center of this group.
    
    Returns:
    
    center of this group
  - conjugate
```
public PermutationGroup conjugate(Permutation permutation)
```
    Returns the conjugate permutation group of this with the specified permutation (this ^ permutation).
    
    Parameters:
    
    permutation - some permutation
    
    Returns:
    
    conjugate permutation group of this with the specified permutation
  - equals
```
public boolean equals(java.lang.Object obj)
```
    Returns true if specified group is equals to this group, i.e. it is isomorphic and acts same on the Ω(degree).
    
    Overrides:
    
    equals in class java.lang.Object
    
    Parameters:
    
    obj - permutation group
    
    Returns:
    
    true if specified group has the same order and all its generators are contained in this group
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object
  - toStringJava
```
public java.lang.String toStringJava()
```
    Returns string to directly paste into Java code.
    
    Returns:
    
    Java code for this permutation group

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Summary:
Nested |
Field |
Constr |
Method

Detail:
Field |
Constr |
Method

Class PermutationGroup

Field Summary

Method Summary

Methods inherited from class java.lang.Object

Methods inherited from interface java.lang.Iterable

Field Detail

TRIVIAL_GROUP

Method Detail

trivialGroup

createPermutationGroup

createPermutationGroup

createPermutationGroupFromBSGS

symmetricGroup

antisymmetricGroup

alternatingGroup

getPositionsInOrbits

generators

degree

orbit

orbitSize

orbit

orbits

indexOfOrbit

isTransitive

isTransitive

isTrivial

isAbelian

randomSource

union

union

getBSGS

getBase

order

ordering

membershipTest

membershipTest

randomElement

randomElement

getBSGSCandidate

isSymmetric

isAlternating

isRegular

pointwiseStabilizer

pointwiseStabilizerRestricted

normalClosureOf

commutator

derivedSubgroup

setwiseStabilizer

containsSubgroup

leftCosetRepresentatives

rightCosetRepresentatives

leftTransversalOf

union

intersection

directProduct

mapping

mapping

iterator

centralizerOf

centralizerOf

center

conjugate

equals

toString

toStringJava