The element order sum, often denoted $\mathrm{\psi }\left(G\right)$, of a finite group $G$, is the sum of the orders of all the elements of $G$. Note that this is an odd positive integer.

The ElementOrderSum( G ) command computes the class element order sum of a finite group G.

The MaximumElementOrder( G ) command returns the largest order of an element of the finite group G.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

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

$31$

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

$3$

$p≔\mathrm{OrderClassPolynomial}\left(G\,x\right)$

$p≔8x^3+3x^2+x$

$\mathrm{eval}\left(\mathrm{diff}\left(p\,x\right)\,x=1\right)$

$31$

$\mathrm{degree}\left(p\,x\right)$

$3$

$\mathrm{SearchSmallGroups}\left(\frac{'\mathrm{elementordersum}'}{'\mathrm{order}'}<\frac{\mathrm{ElementOrderSum}\left(\mathrm{Alt}\left(5\right)\right)}{\mathrm{GroupOrder}\left(\mathrm{Alt}\left(5\right)\right)}\&comma;'\mathrm{soluble}'=\mathrm{false}\right)$

$A≔\mathrm{PerfectGroup}\left(262080\&comma;1\right)\&colon;$

$\mathrm{IsSimple}\left(A\right)$

$\mathrm{true}$

$\mathrm{ElementOrderSum}\left(A\right)$

$12106687$

$B≔\mathrm{PerfectGroup}\left(262080\&comma;2\right)\&colon;$

$\mathrm{IsSimple}\left(B\right)$

$\mathrm{false}$

$\mathrm{ElementOrderSum}\left(B\right)$

$10547861$

The GroupTheory[ElementOrderSum] and GroupTheory[MaximumElementOrder] commands were introduced in Maple 2020.

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