OrderClassNumber - Maple Help

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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
 OrderRank
 compute the order rank of a finite group

 Calling Sequence OrderClassProfile( G, opts ) OrderClassPolynomial( G, x ) OrderClassNumber( G ) OrderRank( 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.
 • The order rank of a finite group $G$ is the number of distinct order class lengths of $G$ greater than $1$.
 • The OrderRank( G ) command returns the order rank of the finite group G.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{OrderClassProfile}\left(G\right)$
 $\left[\left[{1}{,}{1}\right]{,}\left[{2}{,}{3}\right]{,}\left[{3}{,}{8}\right]\right]$ (2)
 > $\mathrm{OrderClassProfile}\left(G,'\mathrm{output}'="list"\right)$
 $\left[{1}{,}{2}{,}{2}{,}{2}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}\right]$ (3)
 > $\mathrm{OrderClassProfile}\left(G,'\mathrm{output}'="multiset"\right)$
 $\left\{\left[{1}{,}{1}\right]{,}\left[{2}{,}{3}\right]{,}\left[{3}{,}{8}\right]\right\}$ (4)
 > $\mathrm{OrderClassNumber}\left(G\right)$
 ${3}$ (5)
 > $\mathrm{OrderRank}\left(G\right)$
 ${2}$ (6)
 > $\mathrm{OrderClassPolynomial}\left(\mathrm{Symm}\left(6\right),'x'\right)$
 ${240}{}{{x}}^{{6}}{+}{144}{}{{x}}^{{5}}{+}{180}{}{{x}}^{{4}}{+}{80}{}{{x}}^{{3}}{+}{75}{}{{x}}^{{2}}{+}{x}$ (7)

Compatibility

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