 Degree - Maple Help

GroupTheory

 Degree
 return the degree of a permutation group
 Support
 return the set of points moved by a permutation group
 SupportLength
 return the number of points moved by a permutation group
 MinSupport
 return the smallest point moved by a permutation group
 MaxSupport
 return the largest point moved by a permutation group Calling Sequence Degree( G ) Support( G ) SupportLength( G ) MinSupport( G ) MaxSupport( G ) Parameters

 G - a permutation group Description

 • The degree of a permutation group is the cardinality of the set upon which it acts.  Since permutation groups act on sets of the form $\left\{1,2,\dots ,n\right\}$, the degree is the positive integer $n$. In other words, the degree $n$ is the smallest positive integer such that $G$ is a subgroup of ${\mathbf{S}}_{n}$ .
 • The Degree( G ) command returns the degree of a permutation group G.
 • Some group constructors, such as those for the symmetric and alternating groups take a degree parameter as input, and the Degree command returns this value.
 • The support of the permutation group $G$ is the set of positive integers displaced by some member of $G$.
 • The Support( G ) command returns the support of G as a Maple set of positive integers.
 • The SupportLength( G ) command returns the cardinality of the support of G. Note that this may differ from the degree of G.
 • The MinSupport( G ) command returns the smallest positive integer that is displaced by the permutation group G. Similarly, the MaxSupport( G ) command returns the largest positive integer displaced by some member of G. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,3,4\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({2}{,}{3}{,}{4}\right)⟩$ (1)
 > $\mathrm{Degree}\left(G\right)$
 ${4}$ (2)
 > $\mathrm{Support}\left(G\right)$
 $\left\{{1}{,}{2}{,}{3}{,}{4}\right\}$ (3)
 > $\mathrm{SupportLength}\left(G\right)$
 ${4}$ (4)
 > $\mathrm{MinSupport}\left(G\right)$
 ${1}$ (5)
 > $\mathrm{MaxSupport}\left(G\right)$
 ${4}$ (6)
 > $G≔\mathrm{DihedralGroup}\left(14\right)$
 ${G}{≔}{{\mathbf{D}}}_{{14}}$ (7)
 > $\mathrm{Degree}\left(G\right)$
 ${14}$ (8)
 > $\mathrm{Support}\left(G\right)$
 $\left\{{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}{,}{9}{,}{10}{,}{11}{,}{12}{,}{13}{,}{14}\right\}$ (9)
 > $\mathrm{SupportLength}\left(G\right)$
 ${14}$ (10)
 > $\mathrm{MinSupport}\left(G\right)$
 ${1}$ (11)
 > $\mathrm{MaxSupport}\left(G\right)$
 ${14}$ (12)
 > $G≔\mathrm{AlternatingGroup}\left(10\right)$
 ${G}{≔}{{\mathbf{A}}}_{{10}}$ (13)
 > $\mathrm{Degree}\left(G\right)$
 ${10}$ (14)
 > $\mathrm{Support}\left(G\right)$
 $\left\{{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}{,}{9}{,}{10}\right\}$ (15)
 > $\mathrm{SupportLength}\left(G\right)$
 ${10}$ (16)
 > $\mathrm{MinSupport}\left(G\right)$
 ${1}$ (17)
 > $\mathrm{MaxSupport}\left(G\right)$
 ${10}$ (18)
 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[2,4,7\right]\right]\right),\mathrm{Perm}\left(\left[\left[4,7\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({4}{,}{7}\right){,}\left({2}{,}{4}{,}{7}\right)⟩$ (19)
 > $\mathrm{Degree}\left(G\right)$
 ${7}$ (20)
 > $\mathrm{Support}\left(G\right)$
 $\left\{{2}{,}{4}{,}{7}\right\}$ (21)
 > $\mathrm{SupportLength}\left(G\right)$
 ${3}$ (22)
 > $\mathrm{MinSupport}\left(G\right)$
 ${2}$ (23)
 > $\mathrm{MaxSupport}\left(G\right)$
 ${7}$ (24) Compatibility

 • The GroupTheory[Degree], GroupTheory[Support], GroupTheory[SupportLength], GroupTheory[MinSupport] and GroupTheory[MaxSupport] commands were introduced in Maple 17.