repthy
Class PGroup

Object
  AbstractCollection
      AbstractSet
          ImmutableSet
              Group
                  HashGroup
                      PGroup

All Implemented Interfaces:

Collection, Set

public class PGroup

extends HashGroup

A p-group, that is, a group of order p^m where p is a prime and m ≥ 0.

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

Author:

Mark McConnell

Field Summary

int	m The order of the group is pm, where p is a prime and m ≥ 0.
int	p The order of the group is pm, where p is a prime and m ≥ 0.

Fields inherited from class Group

charTable

Constructor Summary

PGroup(Group s, int p)
Given a Group and a prime p, this constructs a p-group with the same elements.

Method Summary

protected void	fillInCharTable() Fills in the character table using a special algorithm for p-groups.
Set	getMaximalSubgroups() Returns the set of all maximal subgroups of this p-group, that is, the subgroups of order pm-1.
List	getSubgroups() Returns a List containing every subgroup of this p-group exactly once, including this itself and {1}.
String	orderAsPower() Returns, for example, "2^4" if the order is 16.
String	toString()

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, 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

Methods inherited from class Object

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

Methods inherited from interface Set

containsAll, isEmpty, toArray, toArray

Field Detail

p

public final int p

The order of the group is p^m, where p is a prime and m ≥ 0.

m

public final int m

The order of the group is p^m, where p is a prime and m ≥ 0.

Constructor Detail

PGroup

public PGroup(Group s,
              int p)

Given a Group and a prime p, this constructs a p-group with the same elements.

Throws:

IllegalArgumentException - If this is not a p-group.

Method Detail

orderAsPower

public String orderAsPower()

Returns, for example, "2^4" if the order is 16.

toString

public String toString()

fillInCharTable

protected void fillInCharTable()
                        throws OrthonormalityException

Fills in the character table using a special algorithm for p-groups.

[The Greek letters are wrong in this paragraph.] For any supersolvable group G, Serre's book says every irreducible character chi of G is induced from a monomial character on some subgroup H of G. If M is a maximal subgroup of G containing H, then inducing chi from H to G is the same as inducing (Ind chi from H to M) from M to G. In other words, chi is induced from some character of M, and the latter is a fortiori irreducible. Rose's book shows that for nilpotent groups [check], a maximal subgroup M must be normal in G. Thus to get the irreducible characters of G, it suffices to do two things:

• Get all the degree-1 characters of G. This is the same as to get all the characters of the abelianization G/[G,G] and pull back.
• For each maximal normal M, run through all the irreducible characters of M, induce them up to G, and keep all the results that are irreducible. To run through maximal normal M, run through the non-trivial degree-1 characters from the previous part, and take their kernels.

Overrides:

fillInCharTable in class Group

Throws:

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

getMaximalSubgroups

public Set getMaximalSubgroups()

Returns the set of all maximal subgroups of this p-group, that is, the subgroups of order p^m-1. Each subgroup is an instance of PGroup.

getSubgroups

public List getSubgroups()

Returns a List containing every subgroup of this p-group exactly once, including this itself and {1}. It's the lattice of all subgroups, without the lattice structure. The orders of the subgroups decrease as you go through the list. Each subgroup is an instance of PGroup.