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Class PSL3

Object
  AbstractCollection
      AbstractSet
          ImmutableSet
              Group
                  HashGroup
                      PSL
                          PSL3

All Implemented Interfaces:

Collection, Set

public class PSL3

extends PSL

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

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

Author:

Mark McConnell

Field Summary

Fields inherited from class Group

charTable

Constructor Summary

PSL3(int p)

Method Summary

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 void	main(String[] args) Give it one argument, p.

Methods inherited from class PSL

isSimple

Methods inherited from class HashGroup

contains, getConjClass, getConjClassIndex, getConjClassNum, iterator, 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, 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

PSL3

public PSL3(int p)

Method Detail

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)

Give it one argument, p.