- Transitive Permutation Groups
- Selection Functions
- Interfacing with the library
- Supplemental Downloads

The Transitive Groups Library is created by Alexander Hulpke. You are free to distribute it, provided that you make no modifications.

The actual groups in the library and their generating sets are due to a number of authors: Gregory Butler, John McKay, Gordon Royle, Alexander Hulpke, John Cannon, Gareth Tracy, and Derek Holt.

The list of transitive groups up to degree 11 was published in BM83, the list of degree 12 was published in Roy87, degree 14 and 15 were published in Butler93 and degrees 16--30 were published in Hulpke96 and HulpkeTG. Degree 32 was published in CanHolt32, degrees 34-46 have been obtained by Derek Holt and Gordon Royle in HoltRoyle47, and degree 48 is due to Derek Holt, Gordon Royle, and Gareth Tracy in HoltRoyleTracy48. Groups of prime degree of course are primitive and were known long before.

The transitive groups library will contain representatives for all transitive permutation groups of degree at most 47. Not all degrees might be yet available in the current release. Due to the total number, degree 32 and 48 need to be downloaded separately, see section ``Supplemental Downloads'' below.

Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one. In other words, if they lie in the same conjugacy class under operation of the full symmetric group by conjugation.

`TransitiveGroupsAvailable( `

` ) F`

returns whether the transitive groups groups of degree `deg` are available
for use. This function should be used to test for the scope of the library
available.

`TransitiveGroup( `

`, `

` ) F`

returns the `nr`-th transitive group of degree `deg`. Both `deg` and
`nr` must be positive integers. The transitive groups of equal degree
are sorted with respect to their size, so for example
`TransitiveGroup( `

`deg``, 1 )`

is a transitive group of degree and
size `deg`, e.g, the cyclic group of size `deg`, if `deg` is a
prime.

`NrTransitiveGroups( `

` ) F`

returns the number of transitive groups of degree `deg` stored in the
library of transitive groups. The function returns `fail`

if `deg` is
beyond the range of the library.

The arrangement and the names of the groups of degree up to 15 is the same
as given in ConwayHulpkeMcKay98. With the exception of the symmetric
and alternating group (which are represented as `SymmetricGroup`

and
`AlternatingGroup`

) the generators for these groups also conform to this
paper with the only difference that 0 (which is not permitted in GAP for
permutations to act on) is always replaced by the degree.

The arrangement for all degrees is intended to be equal to the arangement within the system Magma, thus it should be safe to refer to particular (classes of) groups by their index numbers.

gap> TransitiveGroup(10,22); S(5)[x]2 gap> l:=AllTransitiveGroups(NrMovedPoints,12,Size,1440,IsSolvable,false); [ S(6)[x]2, M_10.2(12)=A_6.E_4(12)=[S_6[1/720]{M_10}S_6]2 ] gap> List(l,IsSolvable); [ false, false ]

`TransitiveIdentification( `

` ) A`

Let `G` be a permutation group, acting transitively on a set of up to 30
points. Then `TransitiveIdentification`

will return the position of this
group in the transitive groups library. This means, if `G` acts on
`m` points and `TransitiveIdentification`

returns `n`, then `G` is
permutation isomorphic to the group `TransitiveGroup(m,n)`

.

Note: The points moved do **not** need to be [1..`n`], the group
`langle(2,3,4),(2,3)rangle` is considered to be transitive on 3
points. If the group has several orbits on the points moved by it the
result of `TransitiveIdentification`

is undefined.

gap> TransitiveIdentification(Group((1,2),(1,2,3))); 2

`MinimalTransitiveIndices( `

` ) F`

returns a list of indices of the transitive groups of degree `deg`, which
are minimally trransitive, i.e. for which every proper subgroup is not
transitive.

gap> MinimalTransitiveIndices(12); [ 1, 2, 3, 4, 5, 7, 9, 17, 31, 34, 40, 46, 47, 57, 162, 166, 246 ]

`AllTransitiveGroups( `

`, `

`, ... ) F`

`AllLibraryTransitiveGroups( `

`, `

`, ... ) F`

`OneTransitiveGroup( `

`, `

`, ... ) F`

These functions take an arbitrary number of pairs (but at least one pair) of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group included in the selection, or a list of such values. It returns all (ore one) group satisfying the parameters:

gap> AllTransitiveGroups(NrMovedPoints,[10..15], > Size, [1..100], > IsAbelian, false );

returns a list of all transitive groups with degree between 10 and 15 and size less than 100 that are not abelian.

Thus the `AllTransitiveGroups`

behaves as if it was implemented by a
function similar to the one defined below, where `TransitiveGroupsList`

is a
list of all transitive groups. (Note that in the definition below we assume
for simplicity that `AllTransitiveGroups`

accepts exactly 4 arguments. It is
of course obvious how to change this definition so that the function would
accept a variable number of arguments.)

AllTransitiveGroups := function( fun1, val1, fun2, val2 ) local groups, g, i; groups := []; for i in [ 1 .. Length( TransitiveGroupsList ) ] do g := TransitiveGroupsList[i]; if fun1(g) = val1 or IsList(val1) and fun1(g) in val1 and fun2(g) = val2 or IsList(val2) and fun2(g) in val2 then Add( groups, g ); fi; od; return groups; end;Note that the real selection functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set of properties which are precomputed for the library without having to compute them anew for each group. This will substantially speed up the selection process.

The selection functions for the transitive
groups library are `AllTransitiveGroups`

and `OneTransitiveGroup`

. They
obtain the following properties from the database without having to compute
them anew:

`NrMovedPoints`

, `Size`

, `Transitivity`

, and `IsPrimitive`

.

The function `AllLibraryTransitiveGroups`

works the same way as
`AllTransitiveGroups`

but does not warn if no degree is specified.

Only access using the functions described in this manual is promised to
remain stable. Code that wants to use the transitive groups library should
use `TransitiveGroupsAvailable`

to establish the data being installed for
the desired degree. (This function has a dummy equivalent in the main GAP
library so that it is always available to return `false`

.) Then
`NrTransitiveGroups`

should be used to determine the range of valid indices for
the given degree. Routines should not try to access data structures of the
library directly.

There are almost 3 million groups of degree 32 and these groups require over a GB of disk space when uncompressed. These groups are therefore not part of the package distribution, but are made available as a supplemental download at

https://www.math.colostate.edu/~hulpke/transgrp/trans32.tgz

Simply unpack this archive (which is a tar archive compressed with gzip --
consult your computer administrator on the correct way of unpacking)
in the top folder of the `transgrp`

package (this
folder will contain the `PackageInfo.g`

file, it will create a folder dat32
containing the groups of degree 32. Once you restart GAP it will
automatically recognize the existence of these groups.

The `195826352` groups of degree 48 take over 30GB of disk space in
compressed form and thus require a particular download:

Create a folder `dat48`

in the top folder of the `transgrp`

package. Then
download the 11 files `Trans48Part....tar`

from

https://zenodo.org/record/5935751and unpack them into this folder.

`.gz`

files`.tar`

archive files afterwards.)
Once GAP is restarted these groups will be available, but
`TransitiveIdentification`

(for groups not created from the library) or
the `AllTransitiveGroups`

selector will not work.
Conversion of these groups to GAP is due in part to Jesse Lansdown, whose help is greatly appreciated

Due to the compressed storage, it is perceivable that the groups of degree 48 do not work under Windows. This has not been tested and no promise for this particular functionality is made.

transgrp manual

February 2022