compute the sum of the orders of the elements of a finite group
compute the largest order of an element of a finite group
ElementOrderSum( G )
MaximumElementOrder( G )
a finite group
The element order sum, often denoted ψ⁡G, 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.
G ≔ Alt⁡4
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 ≔ OrderClassPolynomial⁡G,x
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 ≔ PerfectGroup⁡262080,1:
B ≔ PerfectGroup⁡262080,2:
The GroupTheory[ElementOrderSum] and GroupTheory[MaximumElementOrder] commands were introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
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