2 Functionality of the Cubefree package

The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of groups of this size. If possible, the groups are given as pc groups and as permutations groups otherwise.

The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of solvable groups of this size. The groups are given as pc groups.

The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of nilpotent groups of this size. The groups are given as pc groups.

The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of simple groups of this size. In particular, there exists either none or exactly one simple group of the given order.

The input order has to be a positive cubefree integer. The output is a complete and irredundant list of isomorphism type representatives of Frattini-free groups of this size.

Returns an isomorphism between two cubefree groups G and H, if exists, and fail otherwise. It is assumed that the input groups are permutation groups or pc groups. The algorithm is currently efficient only for solvable input groups due to the lack of a constructive recognition algorithm for the simple factors PSL.

Returns true/false, depending on whether two cubefree groups G and H are isomorphic. It is assumed that the input groups are permutation groups or pc groups.

The input n has to be a positive cubefree integer and the output is the number of all cubefree groups of order n. The SmallGroups library is used for squarefree orders, orders of the type p² and p²q, and cubefree orders less than 50000. Only if bool is set to false, then only the squarefree orders and orders of the type p² and p²q,are taken from the SmallGroups library.

The input n has to be a positive cubefree integer and the output is the number of all cubefree solvable groups of order n. The SmallGroups library is used for squarefree orders, orders of the type p² and p²q, and cubefree orders less than 50000. Only if bool is set to false, then only the squarefree orders and orders of the type p² and p²q,are taken from the SmallGroups library.

The input is a positive integer n and the output is a list L of size n such that L[i] contains the number of isomorphism types of groups of order i if i is cubefree and L[i] is not bound, otherwise, 1 ≤ i ≤ n. The SmallGroups library is used for squarefree orders, orders of the type p² and p²q, and cubefree orders less than 50000. Only if bool is set to false, then only the squarefree orders and orders of the type p² and p²q are taken from the SmallGroups library. This function was implemented only for experimental purposes and its implementation could be improved.

This function displays some (very vague) information about the complexity of the construction of the groups of (cubefree) order n. 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: In fact, for each of these pairs (a,b) one would have to construct up to conjugacy all subgroups of order a/b of Aut(S). The sum of the numbers of these subgroups for all pairs (a,b) as above is the number of groups of order n. Thus the output of CubefreeOrderInfo is a trivial lower bound for the number of groups of order n. There is no additional information displayed if bool is set to false.

The input has to be a cubefree integer between 1 and 50000. This function tests the functionality of Cubefree, i.e. 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.

The output is true if n is a cubefree integer and false otherwise.

The output is true if n is a squarefree integer and false otherwise.

The current version of this function allows only n=2. The input q has to be a prime-power q=p^r with p ≥ 5 a prime. The output is a list of all irreducible subgroups of GL(2,q) up to conjugacy.

The input G has to be an absolutely irreducible matrix group over a finite field GF(q). If possible, the output is G rewritten over the subfield of GF(q) generated by the traces of the elements of G. If no rewriting is possible, then the input G is returned.

2.2 Comments on the implementation

This section provides some information about the implementations.

ConstructAllCFGroups

The function ConstructAllCFGroups constructs all groups of a given cubefree order up to isomorphism using the Frattini Extension Method as described in Di05, DiEi05, BeEia, and BeEib. One step in the Frattini Extension Method is to compute Frattini extensions and for this purpose some already implemented methods of the required GAP package GrpConst are used.

Since ConstructAllCFGroups requires only some special types of irreducible subgroups of GL(2,p) (e.g. of cubefree order), it contains a modified internal version of IrreducibleSubgroupsOfGL. This means that the latter is not called explicitely by ConstructAllCFGroups.

ConstructAllCFSimpleGroups and ConstructAllCFNilpotentGroups

The construction of simple or nilpotent groups of cubefree order is rather easy, see Di05 or DiEi05. In particular, the methods used in these cases are independent from the methods used in the general cubefree case.

CountAllCFGroupsUpTo

As described in Di05 and DiEi05, every cubefree group G has the form G=A×I where A is trivial or non-abelian simple and I is solvable. Further, there is a one-to-one correspondence between the solvable cubefree groups and some solvable Frattini-free groups. This one-to-one correspondence allows to count the number of groups of a given cubefree order without computing any Frattini extension. To reduce runtime, the computed irreducible and reducible subgroups of the general linear groups GL(2,p) and also the number of the computed solvable Frattini-free groups are stored during the whole computation. This is very memory consuming but reduces the runtime significantly. The alternative is to run a loop over NumberCFGroups. This function was implemented only for experimental purposes and its implementation could be improved.

IrreducibleSubgroupsOfGL

If the input is a matrix group over GF(q), then the algorithm needs to construct GF(q³) or GF(q⁶) internally.

RewriteAbsolutelyIrreducibleMatrixGroup

The function RewriteAbsolutelyIrreducibleMatrixGroup as described algorithmically in GlHo97 is a probabilistic Las Vegas algorithm; it retries until a correct answer is returned. If the input is G ≤ GL(d,p^r), then the expected runtime is O(rd³).

2.3 Comments on the efficiency

The package GrpConst contains several implementations of algorithms to construct groups of a given order. One of these algorithms is the Frattini extension method, see Chapter 1. The algorithm used in Cubefree is a modification of the Frattini extension method to the case of cubefree orders.

The advantage of this modification is that the isomorphism problem at the construction of Frattini extensions is solved completely on a theoretic level. Also, the construction of the Frattini-free groups up to isomorphism is reduced to the determination of certain subgroups of groups of the type GL(2,p) and C_p−1, p a prime, and to the construction of subdirect products of these subgroups. As this is exponential, this is a main bottleneck of the current implementation.

A modification of the Frattini extension method to squarefree orders yields a powerful construction algorithm for squarefree groups which is based on number theory only. An implementation of this algorithm can be found in the SmallGroups library. Thus for squarefree groups one should definitely use AllSmallGroups and NumberSmallGroups instead of the functions of Cubefree. The same holds for groups of order p² or p²q.

Moreover, using the functionality of Cubefree, the SmallGroups library now contains all groups of cubefree order at most 50000. Hence, also in this case, one should prefer AllSmallGroups and NumberSmallGroups to access the data of the library directly.

For all other cubefree orders n one can try to use Cubefree to construct or count the corresponding groups. Note, that the success of these computations depends basically on the complexity and number theory of the prime-power factorization of n. For each prime p with p² | n one might have to construct subgroups of GL(2,p) and subdirect products involving these subgroups. One can use the info class InfoCF to get some information during the computation. In order to construct subdirect products, we need a permutation representation of these matrix groups. To rewrite them at once, we compute a permutation representation of GL(2,p) and apply this isomorphism to the constructed subgroups. Unfortunately, this is quite time and memory consuming for bigger primes.

In other words, Cubefree can note handle unreasonable cubefree orders. To get a rough idea of the complexity of the computation of groups of order n and to get a trivial lower bound for the number of groups, one can use CubefreeOrderInfo(n).

At the end of this section we consider the quotient q(n) of NumberSmallGroups(n) and CubefreeOrderInfo(n) for cubefree 1 ≤ n ≤ 50000. Although for most of these integers we have a small quotient, note that q(n) seems to be unbounded in general. There are 41597 cubefree integers between 1 and 50000 and 26414 of these integers fulfill q(n)=1. Moreover, 13065 of these integers fulfill 1 < q(n) < 5 and the remaining 2118 integers have 5 ≤ q(n) ≤ 54; e.g. n=2²·3·5·7²·13 has q(n)=1221/23.

2.4 An example session

In this section we outline some examples of applications of the methods described above. We included runtimes for all examples, but omitted the output in some cases, since it would be too long to be printed. The runtimes have been obtained on an Intel(R) Pentium(R) 4 CPU 3.00GHz PC running under Linux.

gap> n:=5^2*7*13^2*67^2*97*107;
1377938614325
gap> CubefreeOrderInfo(n,false);
12
gap> Length(ConstructAllCFGroups(n));time;
12
53111

gap> n:=19^2*23^2*29*37*73^2*107^2;
12501895704027377
gap> CubefreeOrderInfo(n,false);
24
gap> NumberCFGroups(n);time;
24
190536
gap> Length(ConstructAllCFGroups(n));time;
24
948319

gap> n:=5^2*13*23^2*43^2*191;
60716861075
gap> CubefreeOrderInfo(n,false);
16
gap> Length(ConstructAllCFGroups(n)); time;
16
29146

Now we compute some more data.

gap>  n:=2*2*3*11*17*67;
150348
gap> CubefreeOrderInfo(n,false);
20
gap> NumberCFGroups(n);time;
145
12073
gap> Length(ConstructAllCFGroups(n)); time;
145
20757
gap> NumberCFSolvableGroups(n);time;
144
11925
gap> Length(ConstructAllCFSolvableGroups(n)); time;
144
18893
gap> Length(ConstructAllCFFrattiniFreeGroups(n)); time;
109
14421
gap> Length(ConstructAllCFNilpotentGroups(n));time;
2
12
gap> Length(ConstructAllCFSimpleGroups(n));time;
1
8

We consider another example with some info class output.

gap> SetInfoLevel(InfoCF,1);
gap> ConstructAllCFGroups(4620);;time;
#I  Construct all groups of order 4620.
#I    Compute solvable Frattini-free groups of order 2310.
#I    Compute solvable Frattini-free groups of order 4620.
#I  Construct 138 Frattini extensions.
#I    Compute solvable Frattini-free groups of order 77.
#I  Construct 1 Frattini extensions.
#I    Compute solvable Frattini-free groups of order 7.
#I  Construct 1 Frattini extensions.
15501

gap> n:=101^2*97*37^2*29^2;
1139236591513
gap> CubefreeOrderInfo(n,false);
8
gap> NumberCFGroups(n);time;
8
36
gap> SetInfoLevel(InfoCF,1);
gap> ConstructAllCFGroups(n);time;
#I  Construct all groups of order 1139236591513.
#I    Compute solvable Frattini-free groups of order 10512181.
#I    Compute solvable Frattini-free groups of order 304853249.
#I    Compute solvable Frattini-free groups of order 388950697.
#I    Compute solvable Frattini-free groups of order 1061730281.
#I    Compute solvable Frattini-free groups of order 11279570213.
#I    Compute solvable Frattini-free groups of order 30790178149.
#I    Compute solvable Frattini-free groups of order 39284020397.
#I    Compute solvable Frattini-free groups of order 1139236591513.
#I  Construct 8 Frattini extensions.
[ <pc group of size 1139236591513 with 7 generators>, 
  <pc group of size 1139236591513 with 7 generators>, 
  <pc group of size 1139236591513 with 7 generators>, 
  <pc group of size 1139236591513 with 7 generators>, 
  <pc group of size 1139236591513 with 7 generators>, 
  <pc group of size 1139236591513 with 7 generators>, 
  <pc group of size 1139236591513 with 7 generators>, 
  <pc group of size 1139236591513 with 7 generators> ]
1848

The last example considers the cubefree order less than 50000 for which the number of groups with this order is maximal: there are 3093 groups of order 44100.

gap> n:=2*2*3*3*5*5*7*7;
44100
gap> CubefreeOrderInfo(n,false);
100
gap> NumberCFSolvableGroups(n,false);time;
3087
572639
gap> Length(ConstructAllCFSolvableGroups(n)); time;
3087
843085
gap> NumberCFGroups(n,false);time;
3093
719245
gap> Length(ConstructAllCFGroups(n)); time;
3093
1016763
gap> Length(ConstructAllCFFrattiniFreeGroups(n)); time;
1305
504451
gap> Length(ConstructAllCFNilpotentGroups(n));time;
16
180

2.5 Accuracy check

We have compared the results of ConstructAllCFGroups with the library of cubefree groups of SmallGroups. Further, we compared the solvable groups constructed by IrreducibleSubgroupsOfGL with the library of IrredSol. We have also done random isomorphism tests to verify that the list of groups we computed is not redundant.

One can call the following test files. The first one constructs some groups of order at most 2000 and compares the results with the SmallGroups library:

RereadPackage("cubefree","tst/testQuick.g");

The command

RereadPackage("cubefree","tst/testBig.g");

constructs the solvable groups of a random cubefree (but not squarefree) order at most 2²⁸−1 and does a random isomorphism test. Depending on the chosen number, the computation might not terminate due to memory problems.

The following constructs the groups of three random cubefree orders less than 50000 compares the result with the SmallGroups library. Depending on the chosen orders, this may take a while:

RereadPackage("cubefree","tst/testSG.g");

The test file testSGlong.g constructs all cubefree groups of order at most 50000 compares the results with the SmallGroups library. There will be a positive progress report every 50th order so that you can abort the test whenever you want.

RereadPackage("cubefree","tst/testSGlong.g");

Three of these four test files use the function CubefreeTestOrder, see Section 2.1.

The last test file compares some results of IrreducibleSubgroupsOfGL with the database of IrredSol. This may take a while:

RereadPackage("cubefree","tst/testMat.g");

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