repthy
Class AbelianGroup

Object
  AbstractCollection
      AbstractSet
          ImmutableSet
              Group
                  HashGroup
                      AbelianGroup

All Implemented Interfaces:

Collection, Set

public class AbelianGroup

extends HashGroup

What's purple and commutes?

An abelian group does not have to be implemented as an AbelianGroup. Any suitable subclass of Group may be used. However, several tools for abelian groups are only available for instances of this class. The AbelianGroup constructor makes it easy to convert a given abelian group into an isomorphic AbelianGroup with the same elements. The method Homomorphism.identity(repthy.Group, repthy.Group) gives the isomorphism.

Author:

Mark McConnell

Field Summary

Fields inherited from class Group

charTable

Constructor Summary

AbelianGroup(Group G)
Given a Group, this constructs an AbelianGroup with the same elements.

Method Summary

protected void	fillInCharTable() Computes the character table directly from the Smith normal form.
Homomorphism	getIsomFromSNF() Return the tautological isomorphism from the Smith normal form of this group (as returned by getSNF()) to this group.
Homomorphism	getIsomToSNF() Return the tautological isomorphism from this group to the Smith normal form of this group (as returned by getSNF()).
Group	getSNF() Returns a group with the same elements as this, but exhibited in Smith normal form with the ProductGroup structure as shown.
Subgroup	getSylow(int p) Returns the unique Sylow p-subgroup of this group, as a Subgroup in which the underlying subgroup is a PGroup.
String	toString() Returns, for example, C_12 X C_2 X C_2.

Methods inherited from class HashGroup

contains, getConjClass, getConjClassIndex, getConjClassNum, iterator, size

Methods inherited from class Group

cyclicSubgpsUpToConj, describeOrder8, elementarySubgps, getCenter, getCentralizer, getCharComparator, getCharTable, getCommutatorSubgroup, getCompositionSeries, getConjClassRep, getEltOrder, getEltOrder, getIdentity, getOrder, getProperNormalSubgroup, 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

Methods inherited from class Object

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

Methods inherited from interface Set

containsAll, isEmpty, toArray, toArray

Constructor Detail

AbelianGroup

public AbelianGroup(Group G)

Given a Group, this constructs an AbelianGroup with the same elements.

Throws:

IllegalArgumentException - If the given group isn't abelian.

Method Detail

getSylow

public Subgroup getSylow(int p)

Returns the unique Sylow p-subgroup of this group, as a Subgroup in which the underlying subgroup is a PGroup.

Overrides:

getSylow in class Group

Throws:

IllegalArgumentException - If p is not prime.

getSNF

public Group getSNF()

Returns a group with the same elements as this, but exhibited in Smith normal form with the ProductGroup structure as shown.

C₁ × (C₂ × (C₃ × ... × C_k))

When k ≥ 2, each C_i is a non-trivial cyclic group (an instance of C_n), and the order of C_i+1 divides the order of C_i.

If the group is cyclic (k = 1), a single C_n is returned.

getIsomFromSNF

public Homomorphism getIsomFromSNF()

Return the tautological isomorphism from the Smith normal form of this group (as returned by getSNF()) to this group.

getIsomToSNF

public Homomorphism getIsomToSNF()

Return the tautological isomorphism from this group to the Smith normal form of this group (as returned by getSNF()).

fillInCharTable

protected void fillInCharTable()
                        throws OrthonormalityException

Computes the character table directly from the Smith normal form.

Overrides:

fillInCharTable in class Group

Throws:

OrthonormalityException - If the table isn't full at the end, or if anything goes wrong in the middle.

toString

public String toString()

Returns, for example, C_12 X C_2 X C_2.