GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/ElementOrderSum

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

Parameters

Description

Examples

Compatibility

Calling Sequence

ElementOrderSum( G )

MaximumElementOrder( G )

Parameters

G

-

a finite group

Description

• 

The element order sum, often denoted ψG, 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

withGroupTheory:

GAlt4

GA4

(1)

ElementOrderSumG

31

(2)

MaximumElementOrderG

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.

pOrderClassPolynomialG,x

p8x3+3x2+x

(4)

evaldiffp,x,x=1

31

(5)

degreep,x

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.

APerfectGroup262080,1:

IsSimpleA

true

(7)

ElementOrderSumA

12106687

(8)

BPerfectGroup262080,2:

IsSimpleB

false

(9)

ElementOrderSumB

10547861

(10)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[ElementOrder]

GroupTheory[OrderClassPolynomial]