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GroupTheory

  

ConjugacyClass

  

construct the conjugacy class of a group element

  

ConjugacyClasses

  

construct all the conjugacy classes of a group

  

ClassNumber

  

count the conjugacy classes of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ConjugacyClass( g, G )

ConjugacyClasses( G )

ClassNumber( G )

Parameters

g

-

an element of the group G

G

-

a group data structure or a character table

Description

• 

The conjugacy class of an element g of a group G is the set of all conjugates gx=x1·g·x for x in G.

• 

The ConjugacyClass( g, G ) command constructs the conjugacy class of an element g of a group G.

• 

The group G must be an instance of a permutation group or a Cayley table group.

• 

The conjugacy class of g is represented as an object cc for which the following methods are defined.

Representative( cc )

returns the representative of the conjugacy class cc

numelems( cc )

returns the number of members of the conjugacy class cc

member( x, cc ) or x in cc

returns true if x belongs to the conjugacy class cc

Elements( cc )

returns the elements of the conjugacy class cc, as a set

• 

The ConjugacyClasses( G ) command computes all the conjugacy classes of a group (or character table) G, and returns them as a set. The group G must be one for which it is possible to compute the set of all elements of G.

• 

The ClassNumber( G ) command computes the number of conjugacy classes of the group (or character table) G.

• 

Note that the class number of a group is a searchable property for the SearchSmallGroups command.

Examples

withGroupTheory:

Conjugacy classes in the symmetric group are determined by the cycle type: the number of cycles of each length. So the conjugacy class in S6 of permutations with one fixed point, one transposition, and one 3-cycle, contains 663 elements: the support of the 3-cycle can be chosen in 63 ways; there are two 3-cycles given the support; and the fixed point can be chosen from the remaining three points in 3 ways. This fixes the transposition. This is verified below.

g1SymmetricGroup6

g1S6

(1)

cConjugacyClass1,2,3,4,5,g1

c1,23,4,5S6

(2)

numelemsc=binomial6,36

120=120

(3)

You can use in to iterate through the members of a conjugacy class.

evalbseqx,xinc=Elementsc

true

(4)

In the following example you iterate through the members of c, and pair them up with their inverses if they occur. In this case, every element is conjugate with its inverse, so all elements are paired up eventually and none are left over.

opsOperationsg1:

singles:

pairs:

forxincdoinverseops:-`/`x;ifinverseinsinglesthensinglessinglesminusinverse;pairsoppairs,inverse,xelsesinglesx,opsinglesendifenddo:

singles

(5)

numelemscnumelemspairs

2

(6)

Since the cycle type of a permutation on n letters corresponds one-to-one with a partition n, the number of different conjugacy classes is equal to the partition function at n.

cs1ConjugacyClassesg1

cs1S6,2,4,35,6S6,1,2S6,1,2,3,4,5,6S6,1,32,54,6S6,1,6,4,23,5S6,1,6,52,4,3S6,2,63,5S6,2,4,3,6,5S6,3,5,6,4S6,3,5,4S6

(7)

numelemscs1=combinat:-numbpart6

11=11

(8)

Examining the conjugacy classes of the quaternion group, given by a Cayley table.

MCayleyTableQuaternionGroup

M1234567823417586341287654123685756873421687523147568413287561243

(9)

g2CayleyTableGroupM

g2 < a Cayley table group with 8 elements >

(10)

cs2ConjugacyClassesg2

cs21 < a Cayley table group with 8 elements > &comma;2 < a Cayley table group with 8 elements > &comma;3 < a Cayley table group with 8 elements > &comma;5 < a Cayley table group with 8 elements > &comma;6 < a Cayley table group with 8 elements >

(11)

mapprint&comma;mapxabsx=numelemsx&comma;cs2&colon;

1 < a Cayley table group with 8 elements > =1

2 < a Cayley table group with 8 elements > =2

3 < a Cayley table group with 8 elements > =1

5 < a Cayley table group with 8 elements > =2

6 < a Cayley table group with 8 elements > =2

(12)

You see that there are two conjugacy classes of size one and three of size two.

There are only two groups of order less than 512 with class number equal to 3.  (In fact, of any finite order.)

cn3SearchSmallGroupsclassnumber=3

cn33&comma;1,6&comma;1

(13)

Verify the class numbers as follows:

map`@`ClassNumber&comma;SmallGroup&comma;cn3

3&comma;3

(14)

The class number of a direct product of groups can be computed if the class numbers of the individual factors are known.

ClassNumberDirectProductBabyMonster&comma;Symm1000&comma;ElementaryGroup19&comma;30

1020347182128879289726571449479665023113345049543549124072606701932879944

(15)

Compatibility

• 

The GroupTheory[ConjugacyClass] and GroupTheory[ConjugacyClasses] commands were introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

• 

The GroupTheory[ClassNumber] command was introduced in Maple 2018.

• 

For more information on Maple 2018 changes, see Updates in Maple 2018.

• 

The GroupTheory[ClassNumber] command was updated in Maple 2020.

See Also

GroupTheory

GroupTheory[Centralizer]

GroupTheory[SearchSmallGroups]