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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{DihedralGroup}\left(5\right)$

$G≔{\mathbf{D}}_5$

$\mathrm{NumInvolutions}\left(G\right)$

$5$

$\mathrm{NumInvolutions}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$

$1$

$\mathrm{NumInvolutions}\left(\mathrm{QuasicyclicGroup}\left(2\right)\right)$

$1$

$\mathrm{NumInvolutions}\left(\mathrm{FrobeniusGroup}\left(21\,1\right)\right)$

$0$

$\mathrm{NumInvolutions}\left(\mathrm{SemiDihedralGroup}\left(n\right)\right)$

$1+2n$

$\mathrm{NumInvolutions}\left(\mathrm{SL}\left(2\,5\right)\right)$

$1$

$\mathrm{NumInvolutions}\left(\mathrm{Symm}\left(30\right)\right)$

$606917269909048575$

$\mathrm{NumInvolutions}\left(\mathrm{Alt}\left(n\right)\right)$

$3\left(\genfrac{}{}{0ex}{}{n}{4}\right)\mathrm{hypergeom}\left(\left[1\,1-\frac{n}{4}\,-\frac{n}{4}+\frac{3}{2}\,-\frac{n}{4}+\frac{5}{4}\,-\frac{n}{4}+\frac{7}{4}\right]\,\left[\frac{3}{2}\,2\right]\,16\right)$

$\mathrm{NumInvolutions}\left(\mathrm{BabyMonster}\left(\right)\right)$

$512299100893413375$

$\mathrm{it}≔\mathrm{AllSmallGroups}\left(12\,'\mathrm{form}'=''permgroup''\,'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\⟨Small\; Groups\; Iterator:\; 12/1\; ..\; 12/5\⟩}$

$G≔\mathrm{DirectProduct}\left(\mathrm{seq}\left(\mathrm{it}\right)\right)\:$

$\mathrm{NumInvolutions}\left(G\right)$

$511$