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GroupTheory

  

OrderClassProfile

  

compute the order profile of the elements of a finite group

  

OrderClassPolynomial

  

compute the order class polynomial of a finite group

  

OrderClassNumber

  

compute the order class number of a finite group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

OrderClassProfile( G, opts )

OrderClassPolynomial( G, x )

OrderClassNumber( G )

Parameters

G

-

a finite group

opts

-

option of the form output = "list", output = "collected" (the default), or output = "multiset"

x

-

name

Description

• 

The order class profile of a finite group G is the sequence of orders of elements of G, including their multiplicities.

• 

The OrderClassProfile( G ) command computes the order class profile of a finite group G. By default, this is returned as a list of pairs of the form [ order, multiplicity ]. The sorted list of element orders can be returned by using the 'output' = "list" option. To produce, instead, a MultiSet, use the 'output' = "multiset" option.

• 

The OrderClassPolynomial( G, x ) command returns a polynomial encoding of the order class data of the finite group G. It is a univariate polynomial in the indeterminate x for which the coefficient of x^k is equal to the number of elements of order k in G.

• 

The order class number of a finite group G is the number of order classes of elements of G.

• 

The OrderClassNumber( G ) command returns the order class number of the finite group G.

Examples

withGroupTheory:

GAlt4

GA4

(1)

OrderClassProfileG

1,1,2,3,3,8

(2)

OrderClassProfileG,output=list

1,2,2,2,3,3,3,3,3,3,3,3

(3)

OrderClassProfileG,output=multiset

1,1,2,3,3,8

(4)

OrderClassNumberG

3

(5)

OrderClassPolynomialSymm6,x

240x6+144x5+180x4+80x3+75x2+x

(6)

Compatibility

• 

The GroupTheory[OrderClassProfile] command was introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

• 

The GroupTheory[OrderClassPolynomial] and GroupTheory[OrderClassNumber] commands were introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

GroupTheory

GroupTheory[ElementOrder]

ListTools[Collect]

MultiSet