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GroupTheory

  

IsAbelianNumber

  

test whether every group of a given order is Abelian

  

IsCyclicNumber

  

test whether every group of a given order is cyclic

  

IsGCLTNumber

  

test whether every group of a given order is a GCLT group

  

IsIntegrableNumber

  

test whether every group of a given order is integrable

  

IsLagrangianNumber

  

test whether every group of a given order is Lagrangian

  

IsMetabelianNumber

  

test whether every group of a given order is metabelian

  

IsMetacyclicNumber

  

test whether every group of a given order is metacyclic

  

IsNilpotentNumber

  

test whether every group of a given order is nilpotent

  

IsOrderedSylowTowerNumber

  

test whether every group of a given order has an ordered Sylow tower

  

IsSimpleNumber

  

test whether a number is the order of a finite simple group

  

IsSolubleNumber

  

test whether every group of a given order is soluble

  

IsSupersolubleNumber

  

test whether every group of a given order is supersoluble

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsAbelianNumber( n )

IsCyclicNumber( n )

IsGCLTNumber( n )

IsIntegrableNumber( n )

IsLagrangianNumber( n )

IsMetabelianNumber( n )

IsMetacyclicNumber( n )

IsNilpotentNumber( n )

IsOrderedSylowTowerNumber( n )

IsSimpleNumber( n , cyclic )

IsSolubleNumber( n )

IsSupersolubleNumber( n )

Parameters

n

-

a positive integer

cyclic

-

(optional) keyword cyclic; use to include prime numbers as simple numbers

Description

• 

This help page describes a selection of number-theoretic commands having group-theoretic significance. These commands describe positive integers n such that each group of order n has some particular property.

• 

A positive integer n is an Abelian number if every group of order n is Abelian. Well-known examples of Abelian numbers include primes and squares of primes. The Abelian numbers are precisely the cube-free nilpotent numbers. They are also the numbers for which every group of order n is isomorphic to the Frattini subgroup of some finite group. The IsAbelianNumber( n ) command returns true if n is an Abelian number, and false otherwise.

• 

A positive integer n is a cyclic number if every group of order n is cyclic. For instance, every prime number is an cyclic number, but so also is 15, which is not prime. Cyclic numbers are easily characterized:  a positive integer n is a cyclic number precisely when it is relatively prime to its (Euler) totient. The IsCyclicNumber( n ) command returns true if n is a cyclic number, and false otherwise.

• 

A positive integer n is a metacyclic number if every group of order n is metacyclic; that is, if it is an extension of a finite cyclic group by another. For example, every square-free number is a metacyclic number, but so too is 45, which is not square-free. On the other hand, every metacyclic number is cube-free since there is a non-metacyclic group of order p3, for each prime number p. The metacyclic numbers were described fully by Pazderski (1959). The IsMetacyclicNumber( n ) command returns true if n is a metacyclic number, and false otherwise.

• 

A metabelian number is a positive integer n for which every group of order n is metabelian; that is, an extension of an Abelian group by another Abelian group. This is equivalent to having an Abelian derived subgroup. The IsMetabelianNumber( n ) command returns true if n is a metabelian number, and false otherwise.

• 

A nilpotent number is a positive integer n such that every group of order n is nilpotent. The nilpotent numbers n are characterized by the condition that, for each pair p,q of distinct prime divisors of n, there is no power pi dividing n such that q divides pi1. The IsNilpotentNumber( n ) command returns true if n is a nilpotent number, and returns false otherwise.

• 

A positive integer n is a Lagrangian number if every group of order n is Lagrangian; that is, if it satisfies the converse of Lagrange's Theorem in the sense that, for each divisor d of n, it has a subgroup of order equal to d. Lagrangian numbers were fully described by Berger (1978). The IsLagrangianNumber( n ) command returns true if n is a Lagrangian number, and false otherwise. (In the literature, Lagrangian groups are most often called "CLT-groups".)

• 

A positive integer n is a GCLT number if every group of order n is a GCLT-group; that is, if it satisfies the following generalized converse of Lagrange's Theorem: for each subgroup H of G, and for each prime divisor p of the index [G:H] of H in G, there is a subgroup L of G containing H such that the index [L:H] of H in L is equal to p. The GCLT-numbers were determined by Jing (2000). The IsGCLTNumber( n ) command returns true if n is a GCLT-number, and false otherwise.

• 

A supersoluble number is a positive integer n such that every group of order n is supersoluble.  The supersoluble numbers were determined by Pazderski, and the determination used in Maple is based upon his results. The IsSupersolubleNumber( n ) command returns true if n is a supersoluble number, and returns false otherwise.

• 

A positive integer n such that every group of order n has an ordered Sylow tower is called an ordered Sylow tower number. The IsOrderedSylowTowerNumber( n ) returns true if n is an ordered Sylow tower number, and false otherwise.

• 

Soluble numbers are those positive integers n for which every group of order n is soluble. For example, by Burnside's Theorem, every positive integer of the form paqb, where p and q are distinct primes, and a and b are positive integers, is a soluble number. Soluble numbers are characterized as those positive integers not divisible by the order of a minimal simple group. The minimal simple groups were determined by Thompson (1968). The IsSolubleNumber( n ) command returns true provided that n is a soluble number, and returns the value false otherwise.

• 

An integrable number is a positive integer n such that every group of order n is "integrable", in the sense that it is isomorphic to the derived subgroup of some finite group.  (Such groups have also been called competent.) The IsIntegrableNumber( n ) command returns true if n is an integrable number, and returns false otherwise.

• 

A simple number is a positive integer n for which a simple group of order n exists. For example, 168 is a simple number because there is a simple group PSL2,7 (or PSL3,2 ) of order 168, while 54 is not a simple number since every group of order 54 is soluble. The IsSimpleNumber( n ) command returns true if n is a simple number, and returns false otherwise. By default, IsSimpleNumber( n ) returns true only if there is a non-Abelian simple group of order n. In particular, by default, it returns false for prime numbers n. Use the cyclic option to include the primes among the simple numbers.

• 

In general, all these commands rely on the ability to factor the integer n.

Examples

withGroupTheory:

All primes are cyclic numbers.

IsCyclicNumber17

true

(1)

There are, however, non-prime cyclic numbers as well.

IsCyclicNumber995

true

(2)

The smallest non-cyclic number is 4.

IsCyclicNumber4

false

(3)

However, as 4 is the square of the prime 2, it is an Abelian number.

IsAbelianNumber4

true

(4)

An example of an Abelian number that is not the square of a prime is 963.

IsAbelianNumber963

true

(5)

The smallest non-Nilpotent number is 6 (the symmetric group of degree 3 is not nilpotent).

IsNilpotentNumber6

false

(6)

However, 6 is a metacyclic number.

IsMetacyclicNumber6

true

(7)

Nilpotent numbers need not be cube-free.

IsNilpotentNumber135

true

(8)

The smallest non-metacyclic number is 8, since the elementary group of order 8 is not metacyclic.

IsMetacyclicNumber8

false

(9)

andmapIsMetacyclicNumber,seq1..7

true

(10)

The smallest non-Lagrangian number is 12; the alternating group on four letters has no subgroup of order 6.

IsLagrangianNumber12

false

(11)

andmapIsLagrangianNumber,seq1..11

true

(12)

It is also the smallest non-supersoluble number.

IsSupersolubleNumber12

false

(13)

(In fact, a finite group is supersoluble if, and only if, each of its subgroups is Lagrangian.)

Every Lagrangian number is a GCLT number, but not conversely.

IsLagrangianNumber18

true

(14)

IsGCLTNumber18

false

(15)

Not every Lagrangian number is an ordered Sylow tower number. The smallest example is 224.

IsLagrangianNumber224

true

(16)

IsOrderedSylowTowerNumber224

false

(17)

andmapIsLagrangianNumberIsOrderedSylowTowerNumber,seq1..223

true

(18)

Conversely, not every ordered Sylow tower number is a Lagrangian number. All three groups of order 75 have an ordered Sylow tower (one of complexion [5, 3]), but the non-abelian group of order 75 is not Lagrangian; it has no subgroup of order 15.

IsOrderedSylowTowerNumber75

true

(19)

IsLagrangianNumber75

false

(20)

This is the smallest example:

andmapIsOrderedSylowTowerNumberIsLagrangianNumber,seq1..74

true

(21)

The number 60 is not a soluble number since there is a non-Abelian simple group (the alternating group of degree 5) of that order.

IsSolubleNumber60

false

(22)

However, 60 is the smallest number that is not a soluble number.

andmapIsSolubleNumber,seq1..59

true

(23)

Because of the existence of a non-abelian simple group of that order, the number 60 is a simple number.

IsSimpleNumber60

true

(24)

There are, in fact, two simple groups of order 20160, so 20160 is a simple number. (It is the smallest number for which there are two simple groups of that order.)

IsSimpleNumber20160

true

(25)

NumSimpleGroups20160

2

(26)

There are no simple groups of order 100, so 100 is not a simple number.

IsSimpleNumber100

false

(27)

By default, the IsSimpleNumber command only returns true for non-prime numbers.

IsSimpleNumber13

false

(28)

To include the Abelian simple groups, use the cyclic option.

IsSimpleNumber13,cyclic

true

(29)

Compatibility

• 

The GroupTheory[IsAbelianNumber], GroupTheory[IsCyclicNumber], GroupTheory[IsGCLTNumber], GroupTheory[IsIntegrableNumber], GroupTheory[IsLagrangianNumber], GroupTheory[IsMetabelianNumber], GroupTheory[IsMetacyclicNumber], GroupTheory[IsNilpotentNumber], GroupTheory[IsOrderedSylowTowerNumber], GroupTheory[IsSolubleNumber] and GroupTheory[IsSupersolubleNumber] commands were introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

• 

The GroupTheory[IsSimpleNumber] command was introduced in Maple 2020.

• 

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

See Also

GroupTheory

GroupTheory[IsAbelian]

GroupTheory[IsCyclic]

GroupTheory[IsLagrangian]

GroupTheory[IsMetabelian]

GroupTheory[IsNilpotent]

GroupTheory[IsSimple]

GroupTheory[IsSoluble]

GroupTheory[IsSupersoluble]

GroupTheory[NumSimpleGroups]

GroupTheory[references]