construct the conjugacy class of a group element
construct all the conjugacy classes of a group
count the conjugacy classes of a group
compute the conjugate rank of a group
ConjugacyClass( g, G )
ConjugacyClasses( G )
ClassNumber( G )
ConjugateRank( G )
an element of the group G
a group data structure or a character table
The conjugacy class of an element g of a group G is the set of all conjugates gx=x−1·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.
The ConjugateRank( G ) command returns the conjugate rank of a finite group (or character table) G, which is the number of distinct lengths of non-trivial (that is, non-central) conjugacy classes of G.
Note that the class number of a group is a searchable property for the SearchSmallGroups command.
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 6⁢63 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.
g1 ≔ SymmetricGroup⁡6
c ≔ ConjugacyClass⁡1,2,3,4,5,g1
You can use in to iterate through the members of a conjugacy class.
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.
ops ≔ Operations⁡g1:
singles ≔ :
pairs ≔ :
forxincdoinverse ≔ ops:-`/`⁡x;ifinverse∈singlesthensingles ≔ singles∖inverse;pairs ≔ op⁡pairs,x,inverseelsesingles ≔ op⁡singles,xend ifend do:
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.
cs1 ≔ ConjugacyClasses⁡g1
Examining the conjugacy classes of the quaternion group, given by a Cayley table.
M ≔ CayleyTable⁡QuaternionGroup⁡
g2 ≔ CayleyTableGroup⁡M
g2≔ < a Cayley table group with 8 elements >
cs2 ≔ ConjugacyClasses⁡g2
cs2≔1 < a Cayley table group with 8 elements > ,2 < a Cayley table group with 8 elements > ,3 < a Cayley table group with 8 elements > ,5 < a Cayley table group with 8 elements > ,6 < a Cayley table group with 8 elements >
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
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.)
cn3 ≔ SearchSmallGroups⁡'classnumber'=3
Verify the class numbers as follows:
The class number of a direct product of groups can be computed if the class numbers of the individual factors are known.
The GroupTheory[ConjugacyClass], GroupTheory[ConjugacyClasses] and GroupTheory[ConjugateRank] 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.
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