GroupTheory/QuasicyclicGroup/CanonicalForm - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : GroupTheory/QuasicyclicGroup/CanonicalForm

GroupTheory[QuasicyclicGroup]

  

CanonicalForm

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

CanonicalForm( g, G, checkopt )

Parameters

g

-

: rational : an element of G

G

-

: QuasicyclicSubgroup : a quasicyclic subgroup

checkopt

-

: identical(check) = truefalse : (optional) option of the form check = t where t is either true (default) or false

Description

• 

Since members of an additive quasicyclic group are represented by ordinary Maple rationals (where a rational q represents its cosetq&plus; of the quotient group&sol; , it follows that rationals differing by an integer represent the same element of a quasicyclic group. Each cosetq&plus; contains an unique non-negative rational number r for which r<1. This rational r is the canonical form of each member of its coset.

• 

Two rationals that belong to an additive quasicyclic group represent the same group element if they have the same canonical form. This is equivalent to their difference being an integer.

• 

For example, in the quasicyclic groupZ3 the elements 49 and 139 represent the same element, as both have

• 

Elements of a multiplicative quasicyclic p-group are the complex p-power roots of unity. These elements have the form &ExponentialE;2Iπmpn, where m and n are non-negative integers, or complex p-power roots of unity in rectangular form, such as −I or 22+I22=12+I22. In addition, products and powers of p-power roots of unity are considered to belong to the multiplicative quasicyclic p-group.

• 

Subject to simplifications performed automatically by the exp function, the canonical form of an element of a multiplicative quasicyclic p-group is an expression of the form &ExponentialE;2Iπmpn, where m is a non-negative integer such that m<pn.

• 

The CanonicalForm( g, G ) method returns the canonical form of the element g of the quasicyclic group G.

• 

If the check = false option is passed, then it is not checked that g is actually an element of G and, if it is not, an incorrect result may be returned. By default, the check option has the value true.

Examples

withGroupTheory&colon;

GQuasicyclicGroup5

G5

(1)

CanonicalForm25&comma;G

25

(2)

CanonicalForm125&comma;G

25

(3)

OperationsG:-`=`25&comma;125

true

(4)

CanonicalForm4&comma;G

0

(5)

CanonicalForm2725&comma;G

225

(6)

CanonicalForm3025&comma;G

15

(7)

evalbCanonicalFormrand25&comma;G&in;seqi25&comma;i&equals;0..24

true

(8)

HSubgroup1625&comma;G

H5

(9)

CanonicalForm25&comma;H

25

(10)

CanonicalForm125&comma;H

25

(11)

CanonicalForm3025&comma;H

15

(12)

gRandomElementH

g1225

(13)

CanonicalFormg&comma;H&equals;CanonicalFormg&comma;G

1225=1225

(14)

GQuasicyclicGroup2&comma;&apos;form&apos;&equals;multiplicative

GC2

(15)

CanonicalFormI&comma;G

I

(16)

g12122&plus;1I2122

g22+I22

(17)

g&in;G

true

(18)

CanonicalFormg6&comma;G

−I

(19)

CanonicalForm&ExponentialE;9IPi16&comma;G

&ExponentialE;9I16π

(20)

CanonicalForm&ExponentialE;27IPi16&ExponentialE;3IPi8&comma;G

&ExponentialE;I16π

(21)

See Also

GroupTheory

GroupTheory[RandomElement]