ElementOrderSum - Maple Help

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GroupTheory

 ElementOrderSum
 compute the sum of the orders of the elements of a finite group
 MaximumElementOrder
 compute the largest order of an element of a finite group

 Calling Sequence ElementOrderSum( G ) MaximumElementOrder( G )

Parameters

 G - a finite group

Description

 • 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 $G$.
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{ElementOrderSum}\left(G\right)$
 ${31}$ (2)
 > $\mathrm{MaximumElementOrder}\left(G\right)$
 ${3}$ (3)

Note that these invariants are encoded within the order class polynomial of a finite group. The element order sum is the result of evaluating the derivative of the order class polynomial at the point $1$, while the maximum element order is the degree of the order class polynomial.

 > $p≔\mathrm{OrderClassPolynomial}\left(G,x\right)$
 ${p}{≔}{8}{}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{+}{x}$ (4)
 > $\genfrac{}{}{0}{}{\frac{\partial }{\partial x}p}{\phantom{x=1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{\partial }{\partial x}p}}{x=1}$
 ${31}$ (5)
 > $\mathrm{degree}\left(p,x\right)$
 ${3}$ (6)

We can demonstrate a counter-example to a 2011 conjecture of Amiri and Amiri that the minimum value of the element order sum of groups whose order is a simple number is that of a simple group. A different counter-example (of the same order) was discovered by Marefat, Iranmanesh and Tehranian in 2013.

 > $A≔\mathrm{PerfectGroup}\left(262080,1\right):$
 > $\mathrm{IsSimple}\left(A\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{ElementOrderSum}\left(A\right)$
 ${12106687}$ (8)
 > $B≔\mathrm{PerfectGroup}\left(262080,2\right):$
 > $\mathrm{IsSimple}\left(B\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{ElementOrderSum}\left(B\right)$
 ${10547861}$ (10)

Compatibility

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