Group Theory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : System : Information : Updates : Maple 2018 : updates/Maple2018/GroupTheory

Group Theory

Several new commands have been added to the GroupTheory package in Maple 2018.

 

New Subgroup Tests

Groups of Prime Power Order

Other New and Updated Commands

New Subgroup Tests

• 

You can test whether a subgroup of a finite group is subnormal or permutable (quasi-normal) by using the new IsSubnormal and IsPermutable commands. These are generalizations of normal subgroups.

with( GroupTheory ):

G := Symm( 4 );

GS4

(1)

H := Subgroup( { Perm( [[1,2],[3,4]] ) }, G );

H1,23,4

(2)

IsSubnormal( H, G );

true

(3)

IsPermutable( H, G );

false

(4)

IsNormal( H, G );

false

(5)

G := Symm( 3 );

GS3

(6)

N := Subgroup( { Perm( [[1,2,3]] ) }, G );

N1,2,3

(7)

IsNormal( N, G );

true

(8)

IsPermutable( N, G );

true

(9)

IsSubnormal( N, G );

true

(10)

Groups of Prime Power Order

• 

To check whether a finite group is a p-group, for a prime number p, use the new IsPGroup command. The PGroupPrime command can be used to return the prime number p for which a group is a finite p-group.

IsPGroup( Symm( 3 ) );

false

(11)

IsPGroup( DihedralGroup( 4 ) );

true

(12)

PGroupPrime( DihedralGroup( 4 ) );

2

(13)

IsPGroup( DirectProduct( CyclicGroup( 128 ), QuaternionGroup() ) );

true

(14)

Other New and Updated Commands

• 

The ElementOrder command has been extended to work on elements of finite finitely presented groups.

G := < a, b | a^2, b^3, (a.b)^5 = 1 >:

ElementOrder( a.b^2 .a, G );

3

(15)
• 

The new ElementPower command computes powers of elements of a permutation group or a Cayley table group.

• 

The new ClassNumber command returns the number of conjugacy classes of a finite group. In some cases, this can be faster than actually computing the conjugacy classes themselves and counting them.

ClassNumber( Symm( 3 ) );

3

(16)

ClassNumber( DirectProduct( Monster(), DihedralGroup( 4 * n ) ) ) assuming n :: posint;

388n+582

(17)
• 

The number of Abelian groups of a given order can be computed by using the new NumAbelianGroups command.

NumAbelianGroups( 1000 );

9

(18)