# Cubefree

Constructing the Groups of a Given Cubefree Order

Version 1.18
Released 2019-09-30

This project is maintained by Heiko Dietrich

# The GAP 4 package Cubefree

## Installation

The installation of the Cubefree package follows standard GAP rules. So the standard method is to unpack the package into the pkg directory of your GAP distribution. This will create an cubefree subdirectory.

## Requirements

The package is written for GAP 4. It requires the packages GrpConst and Polycyclic, and, if the test file tst/testMat.g is called, the package IrredSol.

## Abilities

This package contains an implementation of an algorithm to construct all groups of a (reasonable) given cubefree order up to isomorphism. This algorithm is based on the ideas in [1] and [2] and it is fully described in [3]. The implementation needs a method to construct all irreducible subgroups of GL(2,p) up to conjugacy. We use the method described in [4] for this purpose. In turn, the algorithm of [4] requires a method for writing an irreducible matrix group over a minimal finite field. We use the algorithm described in [5] for this purpose.

The main functions of the package are the following. Please see the documentation for a more detailed description.

1. ConstructAllCFGroups( n ) … constructs all groups of a given cubefree order n.

2. ConstructAllCFSimpleGroups( n ) … constructs all simple groups of a given cubefree order n.

3. ConstructAllCFSolvableGroups( n ) … constructs all solvable groups of a given cubefree order n.

4. ConstructAllCFNilpotentGroups( n ) … constructs all nilpotent groups of a given cubefree order n.

5. ConstructAllCFFrattiniFreeGroups( n ) … constructs all Frattini-free groups of a given cubefree order n.

6. NumberCFGroups( n ) … returns the number of all groups of a given cubefree order n.

7. NumberCFSolvableGroups( n ) … returns the number of all solvable groups of given cubefree order n.

8. CountAllCFGroupUpTo( n ) … counts all cubefree groups of order at most n. The output is a list L whose i.th entry is the number of groups of order i up to isomorphism if i is cube-free and unbound, otherwise.

9. IrreducibleSubgroupsOfGL( 2, q ) … computes all irreducible subgroups of GL(2,q) up to conjugacy where q=p^r is a prime-power with p>=5.

10. RewriteAbsolutelyIrreducibleMatrixGroup( G ) … rewrites an absolutely irreducible subgroup G\leq GL(n,q) over the subfield generated by the traces of the elements of G.

11. CubefreeOrderInfo( n ) … displays some (very vague) information about the complexity of the construction of the groups of (cubefree) order . It returns the number of possible pairs (a,b) where a is the order of a Frattini-free group F with socle S of order b which has to be constructed in order to construct all groups of order n.

12. CubefreeTestOrder( n ) … tests the functionality of the functions (1)–(7) and compares it with the data of the SmallGroups library. It returns true if everything is okay, otherwise an error message will be displayed.

For some input the counting functions use some data of the SmallGroups library. This can be avoided, see the documentation of (6) and (7). Moreover, in the case of squarefree orders and orders of the type p^2 or p^2q it is more practical to use the functions AllSmallGroups and NumberSmallGroups of the SmallGroups library; please see the documentation for more information.

## Test files

There are several test files to call to check the results of the standard methods. The standard test file is tst/testQuick.g. Moreover, there are tst/testSG.g, tst/testSGlong.g, tst/testBig.g and tst/testMat.g. Please see the documentation for more information.

## Contact

Heiko Dietrich heiko.dietrich@monash.edu, Monash University

Issues can also be reported at https://github.com/gap-packages/cubefree/issues.

## References

[1] H. U. Besche and B. Eick. Construction of finite groups, J. Symb. Comput. 27 (1999), 387 – 404.

[2] H. U. Besche and B. Eick. The groups of order at most 1000 except 512 and 768, J. Symb. Comput. 2 (1999), 405 – 413.

[3] H. Dietrich and B. Eick. On The Groups Of Cube-Free Order J. Algebra 292 (2005), 122 – 137

[4] D. L. Flannery and E. A. O`Brien. Linear Groups of small degree over finite fields, Intern. J. Alg. Comput. 15 (2005), 467 – 502

[5] S. P. Glasby and R. B. Howlett. Writing representations over minimal fields, Comm. Alg. 25(6) (1997) 1703 – 1711.