NumImvolutions - Maple Help

GroupTheory

 NumInvolutions
 compute the number of involutions of a group

 Calling Sequence NumInvolutions(G)

Parameters

 G - : Group : a group object

Description

 • An involution of a group $G$ is an element of order equal to $2$. The involutions of a group exert significant control over the structure of the group.
 • Note that a group of odd order has no involutions.
 • The NumInvolutions(G) command computes the number of involutions of the group G, if possible.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DihedralGroup}\left(5\right)$
 ${G}{≔}{{\mathbf{D}}}_{{5}}$ (1)
 > $\mathrm{NumInvolutions}\left(G\right)$
 ${5}$ (2)
 > $\mathrm{NumInvolutions}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$
 ${1}$ (3)
 > $\mathrm{NumInvolutions}\left(\mathrm{QuasicyclicGroup}\left(2\right)\right)$
 ${1}$ (4)
 > $\mathrm{NumInvolutions}\left(\mathrm{FrobeniusGroup}\left(21,1\right)\right)$
 ${0}$ (5)
 > $\mathrm{NumInvolutions}\left(\mathrm{SemiDihedralGroup}\left(n\right)\right)$
 ${1}{+}{2}{}{n}$ (6)
 > $\mathrm{NumInvolutions}\left(\mathrm{SL}\left(2,5\right)\right)$
 ${1}$ (7)
 > $\mathrm{NumInvolutions}\left(\mathrm{Symm}\left(30\right)\right)$
 ${606917269909048575}$ (8)
 > $\mathrm{NumInvolutions}\left(\mathrm{Alt}\left(n\right)\right)$
 ${3}{}\left(\genfrac{}{}{0}{}{{n}}{{4}}\right){}{\mathrm{hypergeom}}{}\left(\left[{1}{,}{1}{-}\frac{{n}}{{4}}{,}{-}\frac{{n}}{{4}}{+}\frac{{3}}{{2}}{,}{-}\frac{{n}}{{4}}{+}\frac{{5}}{{4}}{,}{-}\frac{{n}}{{4}}{+}\frac{{7}}{{4}}\right]{,}\left[\frac{{3}}{{2}}{,}{2}\right]{,}{16}\right)$ (9)
 > $\mathrm{NumInvolutions}\left(\mathrm{BabyMonster}\left(\right)\right)$
 ${512299100893413375}$ (10)
 > $\mathrm{it}≔\mathrm{AllSmallGroups}\left(12,'\mathrm{form}'="permgroup",'\mathrm{output}'="iterator"\right)$
 ${\mathrm{it}}{≔}{\mathrm{⟨Small Groups Iterator: 12/1 .. 12/5⟩}}$ (11)
 > $G≔\mathrm{DirectProduct}\left(\mathrm{seq}\left(\mathrm{it}\right)\right):$
 > $\mathrm{NumInvolutions}\left(G\right)$
 ${511}$ (12)