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.

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

$\mathrm{IsCyclicNumber}\left(17\right)$

$\mathrm{true}$

$\mathrm{IsCyclicNumber}\left(995\right)$

$\mathrm{true}$

$\mathrm{IsCyclicNumber}\left(4\right)$

$\mathrm{false}$

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

$\mathrm{true}$

$\mathrm{IsAbelianNumber}\left(963\right)$

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{IsMetacyclicNumber}\left(6\right)$

$\mathrm{true}$

$\mathrm{IsNilpotentNumber}\left(135\right)$

$\mathrm{true}$

$\mathrm{IsMetacyclicNumber}\left(8\right)$

$\mathrm{false}$

$\mathrm{andmap}\left(\mathrm{IsMetacyclicNumber}\,\left[\mathrm{seq}\right]\left(1..7\right)\right)$

$\mathrm{true}$

$\mathrm{IsLagrangianNumber}\left(12\right)$

$\mathrm{false}$

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

$\mathrm{true}$

$\mathrm{IsSupersolubleNumber}\left(12\right)$

$\mathrm{false}$

$\mathrm{IsLagrangianNumber}\left(18\right)$

$\mathrm{true}$

$\mathrm{IsGCLTNumber}\left(18\right)$

$\mathrm{false}$

$\mathrm{IsLagrangianNumber}\left(224\right)$

$\mathrm{true}$

$\mathrm{IsOrderedSylowTowerNumber}\left(224\right)$

$\mathrm{false}$

$\mathrm{andmap}\left(\mathrm{IsLagrangianNumber}\Rightarrow \mathrm{IsOrderedSylowTowerNumber}\,\left[\mathrm{seq}\right]\left(1..223\right)\right)$

$\mathrm{true}$

$\mathrm{IsOrderedSylowTowerNumber}\left(75\right)$

$\mathrm{true}$

$\mathrm{IsLagrangianNumber}\left(75\right)$

$\mathrm{false}$

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

$\mathrm{true}$

$\mathrm{IsSolubleNumber}\left(60\right)$

$\mathrm{false}$

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

$\mathrm{true}$

$\mathrm{IsSimpleNumber}\left(60\right)$

$\mathrm{true}$

$\mathrm{IsSimpleNumber}\left(20160\right)$

$\mathrm{true}$

$\mathrm{NumSimpleGroups}\left(20160\right)$

$2$

$\mathrm{IsSimpleNumber}\left(100\right)$

$\mathrm{false}$

$\mathrm{IsSimpleNumber}\left(13\right)$

$\mathrm{false}$

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

$\mathrm{true}$

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.