test whether every group of a given order is Abelian
test whether every group of a given order is cyclic
test whether every group of a given order is a GCLT group
test whether every group of a given order is integrable
test whether every group of a given order is Lagrangian
test whether every group of a given order is metabelian
test whether every group of a given order is metacyclic
test whether every group of a given order is nilpotent
test whether every group of a given order has an ordered Sylow tower
test whether a number is the order of a finite simple group
test whether every group of a given order is soluble
test whether every group of a given order is supersoluble
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 )
a positive integer
(optional) keyword cyclic; use to include prime numbers as simple numbers
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 pi−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 pa⁢qb, 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⁡2,7 (or PSL⁡3,2 ) 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.
All primes are cyclic numbers.
There are, however, non-prime cyclic numbers as well.
The smallest non-cyclic number is 4.
However, as 4 is the square of the prime 2, it is an Abelian number.
An example of an Abelian number that is not the square of a prime is 963.
The smallest non-Nilpotent number is 6 (the symmetric group of degree 3 is not nilpotent).
However, 6 is a metacyclic number.
Nilpotent numbers need not be cube-free.
The smallest non-metacyclic number is 8, since the elementary group of order 8 is not metacyclic.
The smallest non-Lagrangian number is 12; the alternating group on four letters has no subgroup of order 6.
It is also the smallest non-supersoluble number.
(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.
Not every Lagrangian number is an ordered Sylow tower number. The smallest example is 224.
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.
This is the smallest example:
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.
However, 60 is the smallest number that is not a soluble number.
Because of the existence of a non-abelian simple group of that order, the number 60 is a simple number.
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.)
There are no simple groups of order 100, so 100 is not a simple number.
By default, the IsSimpleNumber command only returns true for non-prime numbers.
To include the Abelian simple groups, use the cyclic option.
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.
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