repthy
Class PSL3b

Object
  AbstractCollection
      AbstractSet
          ImmutableSet
              Group
                  PSL3b

All Implemented Interfaces:

Collection, Set

public class PSL3b

extends Group

Special topics concerning PSL₃(p) for a prime p. We produce most of the maximal subgroups, following a paper of Mitchell in Trans. AMS, vol. 12 (1911), 207-242.

As in PMatrixModp, we require 2 ≤ p ≤ 127.

This implementation is based on an AbstractSet that enumerates the group afresh every time it's needed. The features for conjugacy classes and character tables are suppressed (throw UnsupportedOperationException). By contrast, PSL3 extends HashGroup and is full-featured, but is more likely to lead to out-of-memory errors.

Author:

Mark McConnell

Field Summary

Fields inherited from class Group

charTable

Constructor Summary

PSL3b(int p)

Method Summary

boolean	contains(Object o)
Set	getConjClass(int i) Returns (a copy of) the i-th conjugacy class.
int	getConjClassIndex(GroupElt g) Returns i such that g is in the i-th conjugacy class.
int	getConjClassNum() Returns the number of conjugacy classes.
Subgroup	getHessianDiv216()
Subgroup	getNonSplitTorus() Returns a non-split torus, where the underlying subgroup is a C_n.
Subgroup	getNormalizerNonSplitTorus() Preserves a triangle in P2(p3), though not in P2(p), making cyclic permutations of the vertices.
Subgroup	getNormalizerSplitTorus() Preserves a triangle in P2(p), making all six permutations of the vertices.
Subgroup	getOrthogonalGroup(byte d) Returns the subgroup preserving the quadratic form {1,1,d} up to scalar multiples.
Subgroup	getParabolic12() The maximal parabolic subgroup fixing the point [1,0,0] in P2(p).
Subgroup	getParabolic21() The maximal parabolic subgroup fixing the line [x,y,0] in P2(p).
static Group	getW(int p)
Iterator	iterator()
static void	main(String[] args) Give it one argument, p.
int	size()

Methods inherited from class Group

cyclicSubgpsUpToConj, describeOrder8, elementarySubgps, fillInCharTable, getCenter, getCentralizer, getCharComparator, getCharTable, getCommutatorSubgroup, getCompositionSeries, getConjClassRep, getEltOrder, getEltOrder, getIdentity, getOrder, getProperNormalSubgroup, getSylow, isAbelian, isCentral, isSimple, isSubgroupOf, order, power, printTest, tableTest

Methods inherited from class ImmutableSet

add, addAll, clear, remove, removeAll, retainAll

Methods inherited from class AbstractSet

equals, hashCode

Methods inherited from class AbstractCollection

containsAll, isEmpty, toArray, toArray, toString

Methods inherited from class Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Methods inherited from interface Set

containsAll, isEmpty, toArray, toArray

Constructor Detail

PSL3b

public PSL3b(int p)

Method Detail

getW

public static Group getW(int p)

size

public int size()

getConjClassNum

public int getConjClassNum()

Description copied from class: Group

Returns the number of conjugacy classes.

Specified by:

getConjClassNum in class Group

getConjClass

public Set getConjClass(int i)

Description copied from class: Group

Returns (a copy of) the i-th conjugacy class. The inverse operation is Group.getConjClassIndex(repthy.GroupElt).

Implementations may choose to return a SortedSet. If they do, code that needs to pick one element out of the conjugacy class will always pick the first. See Group.getConjClassRep(int).

It's recommended that the Set returned be unmodifiable.

Specified by:

getConjClass in class Group

getConjClassIndex

public int getConjClassIndex(GroupElt g)

Description copied from class: Group

Returns i such that g is in the i-th conjugacy class. The inverse operation is Group.getConjClass(int). May throw an exception if the argument is not in the group.

Specified by:

getConjClassIndex in class Group

iterator

public Iterator iterator()

contains

public boolean contains(Object o)

getParabolic12

public Subgroup getParabolic12()

The maximal parabolic subgroup fixing the point [1,0,0] in P²(p).

getParabolic21

public Subgroup getParabolic21()

The maximal parabolic subgroup fixing the line [x,y,0] in P²(p).

getNormalizerSplitTorus

public Subgroup getNormalizerSplitTorus()

Preserves a triangle in P²(p), making all six permutations of the vertices. Equivalently, the normalizer of the diagonal matrices in PSL₃. "Split torus" means the diagonal matrices.

getNonSplitTorus

public Subgroup getNonSplitTorus()

Returns a non-split torus, where the underlying subgroup is a C_n.

getNormalizerNonSplitTorus

public Subgroup getNormalizerNonSplitTorus()

Preserves a triangle in P²(p³), though not in P²(p), making cyclic permutations of the vertices. Equivalently, the normalizer of a non-split torus in PSL₃.

getOrthogonalGroup

public Subgroup getOrthogonalGroup(byte d)

Returns the subgroup preserving the quadratic form {1,1,d} up to scalar multiples. If d is not zero mod p and p > 2, the result should only depend on whether d is a quadratic residue mod p.

getHessianDiv216

public Subgroup getHessianDiv216()

main

public static void main(String[] args)
                 throws OrthonormalityException

Give it one argument, p.

Throws:

OrthonormalityException