IsAbelianNumber - Maple Help

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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 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 ${p}^{3}$, 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 ${p}^{i}$ dividing $n$ such that $q$ divides ${p}^{i}-1$. 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 ${p}^{a}{q}^{b}$, 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 $PSL\left(2,7\right)$ (or $PSL\left(3,2\right)$ ) 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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$

All primes are cyclic numbers.

 > $\mathrm{IsCyclicNumber}\left(17\right)$
 ${\mathrm{true}}$ (1)

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

 > $\mathrm{IsCyclicNumber}\left(995\right)$
 ${\mathrm{true}}$ (2)

The smallest non-cyclic number is $4$.

 > $\mathrm{IsCyclicNumber}\left(4\right)$
 ${\mathrm{false}}$ (3)

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

 > $\mathrm{IsAbelianNumber}\left(4\right)$
 ${\mathrm{true}}$ (4)

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

 > $\mathrm{IsAbelianNumber}\left(963\right)$
 ${\mathrm{true}}$ (5)

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

 > $\mathrm{IsNilpotentNumber}\left(6\right)$
 ${\mathrm{false}}$ (6)

However, $6$ is a metacyclic number.

 > $\mathrm{IsMetacyclicNumber}\left(6\right)$
 ${\mathrm{true}}$ (7)

Nilpotent numbers need not be cube-free.

 > $\mathrm{IsNilpotentNumber}\left(135\right)$
 ${\mathrm{true}}$ (8)

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

 > $\mathrm{IsMetacyclicNumber}\left(8\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{andmap}\left(\mathrm{IsMetacyclicNumber},\left[\mathrm{seq}\right]\left(1..7\right)\right)$
 ${\mathrm{true}}$ (10)

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

 > $\mathrm{IsLagrangianNumber}\left(12\right)$
 ${\mathrm{false}}$ (11)
 > $\mathrm{andmap}\left(\mathrm{IsLagrangianNumber},\left[\mathrm{seq}\right]\left(1..11\right)\right)$
 ${\mathrm{true}}$ (12)

It is also the smallest non-supersoluble number.

 > $\mathrm{IsSupersolubleNumber}\left(12\right)$
 ${\mathrm{false}}$ (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.

 > $\mathrm{IsLagrangianNumber}\left(18\right)$
 ${\mathrm{true}}$ (14)
 > $\mathrm{IsGCLTNumber}\left(18\right)$
 ${\mathrm{false}}$ (15)

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

 > $\mathrm{IsLagrangianNumber}\left(224\right)$
 ${\mathrm{true}}$ (16)
 > $\mathrm{IsOrderedSylowTowerNumber}\left(224\right)$
 ${\mathrm{false}}$ (17)
 > $\mathrm{andmap}\left(\mathrm{IsLagrangianNumber}⇒\mathrm{IsOrderedSylowTowerNumber},\left[\mathrm{seq}\right]\left(1..223\right)\right)$
 ${\mathrm{true}}$ (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 $15$.

 > $\mathrm{IsOrderedSylowTowerNumber}\left(75\right)$
 ${\mathrm{true}}$ (19)
 > $\mathrm{IsLagrangianNumber}\left(75\right)$
 ${\mathrm{false}}$ (20)

This is the smallest example:

 > $\mathrm{andmap}\left(\mathrm{IsOrderedSylowTowerNumber}⇒\mathrm{IsLagrangianNumber},\left[\mathrm{seq}\right]\left(1..74\right)\right)$
 ${\mathrm{true}}$ (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.

 > $\mathrm{IsSolubleNumber}\left(60\right)$
 ${\mathrm{false}}$ (22)

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

 > $\mathrm{andmap}\left(\mathrm{IsSolubleNumber},\left[\mathrm{seq}\right]\left(1..59\right)\right)$
 ${\mathrm{true}}$ (23)

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

 > $\mathrm{IsSimpleNumber}\left(60\right)$
 ${\mathrm{true}}$ (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.)

 > $\mathrm{IsSimpleNumber}\left(20160\right)$
 ${\mathrm{true}}$ (25)
 > $\mathrm{NumSimpleGroups}\left(20160\right)$
 ${2}$ (26)

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

 > $\mathrm{IsSimpleNumber}\left(100\right)$
 ${\mathrm{false}}$ (27)

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

 > $\mathrm{IsSimpleNumber}\left(13\right)$
 ${\mathrm{false}}$ (28)

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

 > $\mathrm{IsSimpleNumber}\left(13,'\mathrm{cyclic}'\right)$
 ${\mathrm{true}}$ (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.