A permutation group is a group of permutations on a set *Ω* of
positive integers (see chapters Groups and Permutations).

Our standard example in this chapter will be the symmetric group of
degree 4, which is defined by the following **GAP3** statements.

gap> s4 := Group( (1,2), (1,2,3,4) ); Group( (1,2), (1,2,3,4) )

This introduction is followed by a section that describes the function that tests whether an object is a permutation group or not (see section IsPermGroup). The next sections describe the functions that are related to the set of points moved by a permutation group (see PermGroupOps.MovedPoints, PermGroupOps.SmallestMovedPoint, PermGroupOps.LargestMovedPoint, and PermGroupOps.NrMovedPoints). The following section describes the concept of stabilizer chains, which are used by most functions for permutation groups (see Stabilizer Chains). The following sections describe the functions that compute or change a stabilizer chain (see StabChain, ExtendStabChain, ReduceStabChain, MakeStabChainStrongGenerators). The next sections describe the functions that extract information from stabilizer chains (see Base for Permutation Groups, ListStabChain, PermGroupOps.Indices, and PermGroupOps.StrongGenerators). The next two sections describe the functions that find elements or subgroups of a permutation group with a property (see PermGroupOps.ElementProperty and PermGroupOps.SubgroupProperty).

If the permutation groups become bigger, computations become slower. In many cases it is preferable then, to use random methods for computation. This is explained in section Random Methods for Permutation Groups.

Because each permutation group is a domain all set theoretic functions can be applied to it (see chapter Domains and Set Functions for Permutation Groups). Also because each permutation group is after all a group all group functions can be applied to it (see chapter Groups and Group Functions for Permutation Groups). Finally each permutation group operates naturally on the positive integers, so all operations functions can be applied (see chapter Operations of Groups and Operations of Permutation Groups). The last section in this chapter describes the representation of permutation groups (see Permutation Group Records).

The external functions are in the file `LIBNAME/"permgrp.g"`

.

- IsPermGroup
- PermGroupOps.MovedPoints
- PermGroupOps.SmallestMovedPoint
- PermGroupOps.LargestMovedPoint
- PermGroupOps.NrMovedPoints
- Stabilizer Chains
- StabChain
- MakeStabChain
- ExtendStabChain
- ReduceStabChain
- MakeStabChainStrongGenerators
- Base for Permutation Groups
- PermGroupOps.Indices
- PermGroupOps.StrongGenerators
- ListStabChain
- PermGroupOps.ElementProperty
- PermGroupOps.SubgroupProperty
- CentralCompositionSeriesPPermGroup
- PermGroupOps.PgGroup
- Set Functions for Permutation Groups
- Group Functions for Permutation Groups
- Operations of Permutation Groups
- Homomorphisms for Permutation Groups
- Random Methods for Permutation Groups
- Permutation Group Records

`IsPermGroup( `

`obj` )

`IsPermGroup`

returns `true`

if the object `obj`, which may be an object
of an arbitrary type, is a permutation group, and `false`

otherwise. It
will signal an error if `obj` is an unbound variable.

gap> s4 := Group( (1,2), (1,2,3,4) );; s4.name := "s4";; gap> IsPermGroup( s4 ); true gap> f := FactorGroup( s4, Subgroup( s4, [(1,2)(3,4),(1,3)(2,4)] ) ); (s4 / Subgroup( s4, [ (1,2)(3,4), (1,3)(2,4) ] )) gap> IsPermGroup( f ); false # see section FactorGroup gap> IsPermGroup( [ 1, 2 ] ); false

`PermGroupOps.MovedPoints( `

`G` )

`PermGroupOps.MovedPoints`

returns the set of moved points of the
permutation group `G`, i.e., points which are moved by at least one
element of `G` (also see PermGroupOps.NrMovedPoints).

gap> s4 := Group( (1,3,5,7), (1,3) );; gap> PermGroupOps.MovedPoints( s4 ); [ 1, 3, 5, 7 ] gap> PermGroupOps.MovedPoints( Group( () ) ); [ ]

`PermGroupOps.SmallestMovedPoint( `

`G` )

`PermGroupOps.SmallestMovedPoint`

returns the smallest positive integer
which is moved by the permutation group `G` (see also
PermGroupOps.LargestMovedPoint). This function signals an error if `G`
is trivial.

gap> s3b := Group( (2,3), (2,3,4) );; gap> PermGroupOps.SmallestMovedPoint( s3b ); 2

`PermGroupOps.LargestMovedPoint( `

`G` )

`PermGroupOps.LargestMovedPoint`

returns the largest positive integer
which is moved by the permutation group `G` (see also
PermGroupOps.SmallestMovedPoint). This function signals an error if
`G` is trivial.

gap> s4 := Group( (1,2,3,4), (1,2) );; gap> PermGroupOps.LargestMovedPoint( s4 ); 4

`PermGroupOps.NrMovedPoints( `

`G` )

`PermGroupOps.NrMovedPoints`

returns the number of moved points of the
permutation group `G`, i.e., points which are moved by at least one
element of `G` (also see PermGroupOps.MovedPoints).

gap> s4 := Group( (1,3,5,7), (1,3) );; gap> PermGroupOps.NrMovedPoints( s4 ); 4 gap> PermGroupOps.NrMovedPoints( Group( () ) ); 0

Most of the algorithms for permutation groups need a **stabilizer chain**
of the group. The concept of stabilizer chains was introduced by Charles
Sims in Sim70.

If *[b _{1}, ..., b_{n}]* is a list of points,

Since *[b _{1}, ..., b_{n}]*, where

Let *R ^{(i)}* be a right transversal of

For permutation groups of small degree this might be possible, but for
permutation groups of large degree it is still not good enough. Our goal
then is to store as few different permutations as possible such that we
can still reconstruct each representative in *R ^{(i)}*, and from them the
elements in

A stabilizer chain (see StabChain, Permutation Group Records) is
stored recursively in **GAP3**. The group record of a permutation group
`G` with a stabilizer chain has the following additional components.

`orbit`

:-

List of orbitpoints of`orbit[1]`

(which is the basepoint) under the action of the generators.

`transversal`

:-

Factorized inverse transversal as defined above.

`stabilizer`

:-

Record for the stabilizer of the point`orbit[1]`

in the group generated by`generators`

. The components of this record are again`generators`

,`orbit`

,`transversal`

,`identity`

and`stabilizer`

. The last stabilizer in the stabilizer chain only contains the components`generators`

, which is an empty list, and`identity`

.

`stabChain`

:-

A record, that contains all information about the stabilizer chain. Functions acessing the stabilizer chain should do it using this record, as it is planned to remove the above three components from the group record in the future. The components of the`stabChain`

record are described in section Permutation Group Records.

Note that the values of these components are changed by functions that change, extend, or reduce a base (see StabChain, ExtendStabChain, and ReduceStabChain).

Note that the records that represent the stabilizers are not group
records (see Group Records). Thus you cannot take such a stabilizer
and apply group functions to it. The last `stabilizer`

in the stabilizer
chain is a record whose component `generators`

is empty.

Below you find an example for a **stabilizer chain** for the symmetric
group of degree 4.

rec( identity := (), generators := [ (1,2), (1,2,3,4) ], orbit := [ 1, 2, 4, 3 ], transversal := [ (), (1,2), (1,2,3,4), (1,2,3,4) ], stabilizer := rec( identity := (), generators := [ (3,4), (2,4) ], orbit := [ 2, 4, 3 ], transversal := [ , (), (3,4), (2,4) ], stabilizer := rec( identity := (), generators := [ (3,4) ], orbit := [ 3, 4 ], transversal := [ ,, (), (3,4) ], stabilizer := rec( identity := (), generators := [ ] ) ) ) )

`StabChain( `

`G` )

`StabChain( `

`G`, `opt` )

`StabChain`

computes and returns a stabilizer chain for `G`. The option
record `opt` can be given and may contain information that will be used
when computing the stabilizer chain.
Giving this information might speed up computations. When using random
methods (see Random Methods for Permutation Groups), `StabChain`

also
guarantees, that the computed stabilizer chain confirms to the information
given. For example giving the size ensures correctness of the stabilizer
chain.

If information of this kind can also be gotten from the parent group,
`StabChain`

does so.

The following components of the option record are currectly supported:

`size`

:

The group size.

`limit`

:

An upper limit for the group size.

`base`

:

A list of points. If given,`StabChain`

computes a reduced base starting with the points in`base`

.

`knownBase`

:

A list of points, representing a known base.

`random`

:

A value to supersede global or parent group setting of`StabChainOptions.random`

(see Random Methods for Permutation Groups).

`MakeStabChain( `

`G` )

`MakeStabChain( `

`G`, `lst` )

`MakeStabChain`

computes a **reduced** stabilizer chain for the permutation
group `G`.

If no stabilizer chain for *G* is already known and no argument `lst` is
given, it computes a reduced stabilizer chain for the lexicographically
smallest reduced base of `G`.

If no stabilizer chain for *G* is already known and an argument `lst` is
given, it computes a **reduced** stabilizer chain with a base that starts
with the points in `lst`. Note that points in `lst` that would lead to
trivial stabilizers will be skipped (see ExtendStabChain).

Deterministically, the stabilizer chain is computed using the
**Schreier-Sims-Algorithm**, which is described in Leo80. The
time used is in practice proportional to the third power of the degree
of the group.

If a stabilizer chain for *G* is already known and no argument `lst` is
given, it reduces the known stabilizer chain.

If a stabilizer chain for *G* is already known and an argument `lst` is
given, it changes the stabilizer chain such that the result is a
**reduced** stabilizer chain with a base that starts with the points in
`lst` (see ExtendStabChain). Note that points in `lst` that would lead
to trivial stabilizers will be skipped.

The algorithm used in this case is called **basechange**, which is
described in But82. The worst cases for the basechange algorithm
are groups of large degree which have a long base.

gap> s4 := Group( (1,2), (1,2,3,4) ); Group( (1,2), (1,2,3,4) ) gap> MakeStabChain( s4 ); # compute a stabilizer chain gap> Base( s4 ); [ 1, 2, 3 ] gap> MakeStabChain( s4, [4,3,2,1] ); # perform a basechange gap> Base( s4 ); [ 4, 3, 2 ]

`MakeStabChain`

mainly works by calling `StabChain`

with appropriate
parameters.

`ExtendStabChain( `

`G`, `lst` )

`ExtendStabChain`

inserts trivial stabilizers into the known stabilizer
chain of the permutation group `G` such that `lst` becomes the base of
`G`. The stabilizer chain which belongs to the base `lst` must reduce to
the old stabilizer chain (see ReduceStabChain).

This function is useful if two different (sub-)groups have to have exactly the same base.

gap> s4 := Group( (1,2), (1,2,3,4) );; gap> MakeStabChain( s4, [3,2,1] ); Base( s4 ); [ 3, 2, 1 ] gap> h := Subgroup( Parent(s4), [(1,2,3,4), (2,4)] ); Subgroup( Group( (1,2), (1,2,3,4) ), [ (1,2,3,4), (2,4) ] ) gap> Base( h ); [ 1, 2 ] gap> MakeStabChain( h, Base( s4 ) ); Base( h ); [ 3, 2 ] gap> ExtendStabChain( h, Base( s4 ) ); Base( h ); [ 3, 2, 1 ]

`ReduceStabChain( `

`G` )

`ReduceStabChain`

removes trivial stabilizers from a known stabilizer
chain of the permutation group `G`. The result is a **reduced** stabilizer
chain (also see ExtendStabChain).

gap> s4 := Group( (1,2), (1,2,3,4) );; gap> Base( s4 ); [ 1, 2, 3 ] gap> ExtendStabChain( s4, [ 1, 2, 3, 4 ] ); Base( s4 ); [ 1, 2, 3, 4 ] gap> PermGroupOps.Indices( s4 ); [ 4, 3, 2, 1 ] gap> ReduceStabChain( s4 ); Base( s4 ); [ 1, 2, 3 ]

`MakeStabChainStrongGenerators( `

`G`, `base`, `stronggens` )

`MakeStabChainStrongGenerators`

computes a **reduced** stabilizer chain for
the permutation group `G` with the base `base` and the strong generating
set `stronggens`. `stronggens` must be a strong generating set for `G`
relative to the base `base`; note that this is not tested. Since the
generators for `G` are not changed the strong generating set of `G` got
by `PermGroupOps.StrongGenerators`

is not exactly `stronggens`
afterwards. This function is mostly used to reconstruct a stabilizer
chain for a group `G` and works considerably faster than `MakeStabChain`

(see MakeStabChain).

gap> G := Group( (1,2), (1,2,3), (4,5) );; gap> Base( G ); [ 1, 2, 4 ] gap> ExtendStabChain( G, [1,2,3,4] ); gap> PermGroupOps.Indices( G ); base := Base( G ); [ 3, 2, 1, 2 ] # note that the stabilizer chain is not reduced [ 1, 2, 3, 4 ] gap> stronggens := PermGroupOps.StrongGenerators( G ); [ (4,5), (2,3), (1,2), (1,2,3) ] gap> H := Group( (1,2), (1,3), (4,5) ); Group( (1,2), (1,3), (4,5) ) # of courseG=Hgap> MakeStabChainStrongGenerators( H, base, stronggens ); gap> PermGroupOps.Indices( H ); Base( H ); [ 3, 2, 2 ] # note that the stabilizer chain is reduced [ 1, 2, 4 ] gap> PermGroupOps.StrongGenerators( H ); [ (4,5), (2,3), (1,2), (1,3) ] # note that this is different from`stronggens`

`Base( `

`G` )

`Base`

returns a base for the permutation group `G`. If a stabilizer
chain for `G` is already known, `Base`

returns the base for this
stabilizer chain. Otherwise a stabilizer chain for the lexicographically
smallest reduced base is computed and its base is returned (see
Stabilizer Chains).

gap> s4 := Group( (1,2,3,4), (1,2) );; gap> Base( s4 ); [ 1, 2, 3 ]

`PermGroupOps.Indices( `

`G` )

`PermGroupOps.Indices`

returns a list `l` of indices of the permutation
group `G` with respect to a stabilizer chain of `G`, i.e.,

is
the index of `l`[`i`]*G ^{(i+1)}* in

`PermGroupOps.Indices`

returns the indices corresponding to this
stabilizer chain. Otherwise a stabilizer chain with the
lexicographically smallest reduced base is computed and the indices
corresponding to this chain are returned (see Stabilizer Chains).

gap> s4 := Group( (1,2,3,4), (1,2) );; gap> PermGroupOps.Indices( s4 ); [ 4, 3, 2 ] # note that for s4 the indices are # actually independent of the base

`PermGroupOps.StrongGenerators( `

`G` )

`PermGroupOps.StrongGenerators`

returns a list of strong generators for
the permutation group `G`. If a stabilizer chain for `G` is already
known, `PermGroupOps.StrongGenerators`

returns a strong generating set
corresponding to this stabilizer chain. Otherwise a stabilizer chain
with the lexicographically smallest reduced base is computed and a strong
generating set corresponding to this chain is returned (see Stabilizer
Chains).

gap> s4 := Group( (1,2,3,4), (1,2) );; gap> Base( s4 ); [ 1, 2, 3 ] gap> PermGroupOps.StrongGenerators( s4 ); [ (3,4), (2,3,4), (1,2), (1,2,3,4) ]

`ListStabChain( `

`G` )

`ListStabChain`

returns a list of stabilizer records of the stabilizer
chain of the permutation group `G`, i.e., the result is a list `l` such
that

is the `l`[`i`]`i`-th stabilizer *G ^{(i)}*. The records in that
list are identical to the records of the stabilizer chain. Thus changes
made in a record

`l`[`i`]

are simultaneously done in the stabilizer
chain (see Identical Records).
`PermGroupOps.ElementProperty( `

`G`, `prop` )

`PermGroupOps.ElementProperty( `

`G`, `prop`, `K` )

`PermGroupOps.ElementProperty`

returns an element `g` in the permutation
group `G` such that

is `prop`(`g`)`true`

. `prop` must be a function
of one argument that returns either `true`

or `false`

when applied to
an element of `G`. If `G` has no such element, `false`

is returned.

gap> V4 := Group((1,2),(3,4));; gap> PermGroupOps.ElementProperty( V4, g -> (1,2)^g = (3,4) ); false

`PermGroupOps.ElementProperty`

first computes a stabilizer chain for `G`,
if necessary. Then it performs a backtrack search through `G` for an
element satisfying `prop`, i.e., enumerates all elements of `G` as
described in section Stabilizer Chains, and applies `prop` to each
until one element `g` is found for which

is `prop`(`g`)`true`

. This
algorithm is described in detail in But82.

gap> S8 := Group( (1,2), (1,2,3,4,5,6,7,8) );; S8.name := "S8";; gap> Size( S8 ); 40320 gap> V := Subgroup( S8, [(1,2),(1,2,3),(6,7),(6,7,8)] );; gap> Size( V ); 36 gap> U := V ^ (1,2,3,4)(5,6,7,8);; gap> PermGroupOps.ElementProperty( S8, g -> U ^ g = V ); (1,4,2)(5,6) # another permutation conjugatingUtoV

This search will of course take quite a while if `G` is large, especially
if no element of `G` satisfies `prop`, and therefore all elements of `G`
must be tried.

To speed up the computation you may pass a subgroup `K` of `G` as
optional third argument. This subgroup must preserve `prop` in the sense
that either all elements of a left coset

satisfy `g`*`K``prop` or no
element of

does.
`g`*`K`

In our example above such a subgroup is the normalizer *N _{G}(V)* because

gap> K := Subgroup( S8, V.generators );; gap> PermGroupOps.ElementProperty( S8, g -> U ^ g = V, K ); (1,4,2)(5,6)

In the following example, we use the same subgroup, but with a larger generating system. This speeds up the computation by another factor of 3. Something like this may happen frequently. The reason is too complicated to be explained here.

gap> K2 := Subgroup( S8, Union( V.generators, [(2,3),(7,8)] ) );; gap> K2 = K; true gap> PermGroupOps.ElementProperty( S8, g -> U ^ g = V, K2 ); (1,4,2)(5,6)

Passing the full normalizer speeds up the computation in this example by
another factor of 2. Beware though that in other examples the
computation of the normalizer alone may take longer than calling
`PermGroupOps.ElementProperty`

with only the subgroup itself as argument.

gap> N := Normalizer( S8, V ); Subgroup( S8, [ (1,2), (1,2,3), (6,7), (6,7,8), (2,3), (7,8), (1,6)(2,7)(3,8), (4,5) ] ) gap> Size( N ); 144 gap> PermGroupOps.ElementProperty( S8, g -> U ^ g = V, N ); (1,4)(5,6)

`PermGroupOps.SubgroupProperty( `

`G`, `prop` )

`PermGroupOps.SubgroupProperty( `

`G`, `prop`, `K` )

`PermGroupOps.SubgroupProperty`

returns the subgroup `U` of the
permutation group `G` of all elements in `G` that satisfy `prop`, i.e.,
the subgroup of all elements `g` in `G` such that

is
`prop`(`g`)`true`

. `prop` must be a function of one argument that returns either
`true`

or `false`

when applied to an element of `G`. Of course the
elements that satisfy `prop` must form a subgroup of `G`.
`PermGroupOps.SubgroupProperty`

builds a stabilizer chain for `U`.

gap> S8 := Group( (1,2), (1,2,3,4,5,6,7,8) );; S8.name := "S8";; gap> Size(S8); 40320 gap> V := Subgroup( S8, [(1,2),(1,2,3),(6,7),(6,7,8)] );; gap> Size(V); 36 gap> PermGroupOps.SubgroupProperty( S8, g -> V ^ g = V ); Subgroup( S8, [ (7,8), (6,7), (4,5), (2,3)(4,5)(6,8,7), (1,2), (1,6,3,8)(2,7) ] ) # the normalizer of`V`

in`S8`

`PermGroupOps.SubgroupProperty`

first computes a stabilizer chain for
`G`, if necessary. Then it performs a backtrack search through `G` for
the elements satisfying `prop`, i.e., enumerates all elements of `G` as
described in section Stabilizer Chains, and applies `prop` to each,
adding elements for which

is `prop`(`g`)`true`

to the subgroup `U`.
Once `U` has become non-trivial, it is used to eliminate whole cosets of
stabilizers in the stabilizer chain of `G` if they cannot contain
elements with the property `prop` that are not already in `U`. This
algorithm is described in detail in But82.

This search will of course take quite a while if `G` is large. To speed
up the computation you may pass a subgroup `K` of `U` as optional third
argument.

Passing the subgroup *V* itself, speeds up the computation in this
example by a factor of 2.

gap> K := Subgroup( S8, V.generators );; gap> PermGroupOps.SubgroupProperty( S8, g -> V ^ g = V, K ); Subgroup( S8, [ (1,2), (1,2,3), (6,7), (6,7,8), (2,3), (7,8), (4,5), (1,6,3,8)(2,7) ] )

`CentralCompositionSeriesPPermGroup( `

`G` )

This function calculates a central composition series for the *p*-group
`G`. The method used is known as Holt's algorithm. If `G` is not a
`p`-group, an error is signalled.

gap> D := Group( (1,2,3,4), (1,3) );; D.name := "d8";; gap> CentralCompositionSeriesPPermGroup( D ); [ d8, Subgroup( d8, [ (2,4), (1,3) ] ), Subgroup( d8, [ (1,3)(2,4) ] ), Subgroup( d8, [ ] ) ]

`PermGroupOps.PgGroup( `

`G` )

This function converts a permutation group `G` of prime power order *p ^{d}*
into an ag group

`CentralCompositionSeriesPPermGroup`

(see
CentralCompositionSeriesPPermGroup). An isomorphism from the ag group
to the permutation group is bound to `P`.bijection

.
There is no dispatcher to this function, it must be called as
`PermGroupOps.PgGroup`

.

All set theoretic functions described in chapter Domains are also applicable to permutation groups. This section describes which functions are implemented specially for permutation groups. Functions not mentioned here are handled by the default methods described in the respective sections.

To compute a random element in a permutation group `G` **GAP3** computes a
stabilizer chain for `G`, takes on each level a random representative and
returns the product of those. All elements of `G` are chosen with equal
probability by this method.

`Size`

calls `StabChain`

(see StabChain), if necessary, and
returns the product of the indices of the stabilizer chain (see
Stabilizer Chains).

`Elements`

calls `StabChain`

(see StabChain), if necessary, and
enumerates the elements of `G` as described in Stabilizer Chains. It
returns the set of those elements.

`Intersection`

first computes stabilizer chains for `G1` and `G2` for a
common base. If either group already has a stabilizer chain a basechange
is performed (see MakeStabChain). `Intersection`

enumerates the
elements of `G1` and `G2` using a backtrack algorithm, eliminating whole
cosets of stabilizers in the stabilizer chains if possible (see
PermGroupOps.SubgroupProperty). It builds a stabilizer chain for the
intersection.

All group functions for groups described in chapter Group are also applicable to permutation groups. This section describes which functions are implemented specially for permutation groups. Functions not mentioned here are handled by the default methods described in the respective sections.

`G` ^ `p`

`ConjugateSubgroup( `

`G`, `p` )

Returns the conjugate permutation group of `G` with the permutation `p`.
`p` must be an element of the parent group of `G`. If a stabilizer chain
for `G` is already known, it is also conjugated.

`Centralizer( `

`G`, `U` )

`Centralizer( `

`G`, `g` )

`Normalizer( `

`G`, `U` )

These functions first compute a stabilizer chain for `G`. If a
stabilizer chain is already known a basechange may be performed to obtain
a base that is better suited for the problem. These functions then
enumerate the elements of `G` with a backtrack algorithm, eliminating
whole cosets of stabilizers in the stabilizer chain if possible (see
PermGroupOps.SubgroupProperty). They build a stabilizer chain for the
resulting subgroup.

If `G` is not transitive, its `p`-Sylow subgroup is computed by starting
with

, and for each transitive constituent homomorphism `P`:=`G``hom`
iterating

.
`P` := PreImage( SylowSubgroup( Image( `hom`, `P` ), `p` ) )

If `G` is transitive but not primitive, its `p`-Sylow subgroup is
computed as

`SylowSubgroup( PreImage( SylowSubgroup(Image(`

.
`hom`,`G`),`p`) ), `p` )

If `G` is primitive, `SylowSubgroup`

takes random elements in `G`, until
it finds a `p`-element `g`, whose centralizer in `G` contains the whole
`p`-Sylow subgroup. Such an element must exist, because a `p`-group has
a nontrivial centre. Then the `p`-Sylow subgroup of the centralizer is
computed and returned. Note that the centralizer must be a proper
subgroup of `G`, because it operates imprimitively on the cycles of `g`.

Returns the coset

. The representative chosen is the
lexicographically smallest element of that coset. It is computed with an
algorithm that is very similar to the backtrack algorithm.
`U`*`g`

gap> s4 := Group( (1,2,3,4), (1,2) );; s4.name := "s4";; gap> u := Subgroup( s4, [(1,2,3)] );; gap> Coset( u, (1,3,2) ); (Subgroup( s4, [ (1,2,3) ] )*()) gap> Coset( u, (3,2) ); (Subgroup( s4, [ (1,2,3) ] )*(2,3))

Returns the cosets of `U` in `G`. `Cosets`

first computes stabilizer
chains for `G` and `U` with a common base. If either subgroup already
has a stabilizer chain, a basechange is performed (see MakeStabChain).
A transversal is computed recursively using the fact that if *S* is a
transversal of *U ^{(2)} = Stab_{U}(b_{1})* in

gap> Cosets( s4, u ); [ (Subgroup( s4, [ (1,2,3) ] )*()), (Subgroup( s4, [ (1,2,3) ] )*(3,4)), (Subgroup( s4, [ (1,2,3) ] )*(2,3)), (Subgroup( s4, [ (1,2,3) ] )*(2,3,4)), (Subgroup( s4, [ (1,2,3) ] )*(2,4,3)), (Subgroup( s4, [ (1,2,3) ] )*(2,4)), (Subgroup( s4, [ (1,2,3) ] )*(1,2,3,4)), (Subgroup( s4, [ (1,2,3) ] )*(1,2,4)) ]

Computes the character of the natural permutation representation of `P`,
i.e. it does the same as `PermutationCharacter( `

but works much faster.
`P`, * Stab_{P}(1)* )

gap> G := SymmetricPermGroup(5);; gap> PermutationCharacter(G); [ 5, 3, 1, 2, 0, 1, 0 ]

This function builds an elementary abelian series of `G` by iterated
construction of normal closures. If a partial elementary abelian series
reaches up to a subgroup `U` of `G` which does not yet contain the
generator `s` of `G` then the series is extended up to the normal closure
`N` of `U` and `s`. If the factor

is not elementary abelian,
i.e., if some commutator of `N`/`U``s` with one of its conjugates under `G` does
not lie in `U`, intermediate subgroups are calculated recursively by
extending `U` with that commutator. If `G` is solvable this process must
come to an end since commutators of arbitrary depth cannot exist in
solvable groups.

Hence this method gives an elementary abelian series if `G` is solvable
and gives an infinite recursion if it is not. For permutation groups,
however, there is a bound on the derived length that depends only on the
degree `d` of the group. According to Dixon this is *(5 *log* _{3}(d))/2*. So
if the commutators get deeper than this bound the algorithm stops and
sets

`G`.isSolvable

to `false`

, signalling an error. Otherwise
`G`.isSolvable

is set to `true`

and the elementary abelian series is
returned as a list of subgroups of

gap> S := Group( (1,2,3,4), (1,2) );; S.name := "s4";; gap> ElementaryAbelianSeries( S ); [ Subgroup( s4, [ (1,2), (1,3,2), (1,4)(2,3), (1,2)(3,4) ] ), Subgroup( s4, [ (1,3,2), (1,4)(2,3), (1,2)(3,4) ] ), Subgroup( s4, [ (1,4)(2,3), (1,2)(3,4) ] ), Subgroup( s4, [ ] ) ] gap> A := Group( (1,2,3), (3,4,5) );; gap> ElementaryAbelianSeries( A ); Error, <G> must be solvable

Solvability of a permutation group `G` is tested by trying to construct
an elementary abelian series as described above. After this has been
done the flag

is set correctly, so its value is
returned.
`G`.isSolvable

gap> S := Group( (1,2,3,4), (1,2) );; gap> IsSolvable( S ); true gap> A := Group( (1,2,3), (3,4,5) );; gap> IsSolvable( A ); false

A composition series for the solvable group `G` is calculated either from
a given subnormal series, which must be bound to

,
in which case `G`.subnormalSeries

must hold the corresponding base-strong
subnormal generating system, or from an elementary abelian series (as
computed by `G`.bssgs`ElementaryAbelianSeries( `

above) by inserting
intermediate subgroups (i.e. powers of the polycyclic generators or
composition series along bases of the vector spaces in the elementary
abelian series). In either case, after execution of this function,
`G` )

holds a base-strong pag system corresponding to the
composition series calculated.
`G`.bssgs

gap> S := Group( (1,2,3,4), (1,2) );; S.name := "s4";; gap> CompositionSeries( S ); [ Subgroup( s4, [ (1,2), (1,3,2), (1,4)(2,3), (1,2)(3,4) ] ), Subgroup( s4, [ (1,3,2), (1,4)(2,3), (1,2)(3,4) ] ), Subgroup( s4, [ (1,4)(2,3), (1,2)(3,4) ] ), Subgroup( s4, [ (1,2)(3,4) ] ), Subgroup( s4, [ ] ) ]

If `G` is not solvable then a composition series `cs` is computed with an
algorithm by A. Seress and R. Beals. In this case the factor group of
each element

in the composition series modulo the next one
`cs`[`i`]

are represented as primitive permutation groups. One
should call `cs`[`i`+1]

directly to avoid the check in `cs`[`i`].operations.FactorGroup( `cs`[`i`], `cs`[`i`+1] )`FactorGroup`

that

is normal
in `cs`[`i`+1]

. The natural homomorphism of `cs`[`i`]

onto this factor
group will be given as a `cs`[`i`]`GroupHomomorphismByImages`

(see
GroupHomomorphismByImages).

gap> pyl29 := Group( (1,2,3)(4,5,6)(7,8,9), (2,6,4,9,3,8,7,5), > (4,7)(5,8)(6,9), (1,10)(4,7)(5,6)(8,9) );; gap> pyl29.name := "pyl29";; gap> cs := CompositionSeries( pyl29 ); [ Subgroup( pyl29, [ (1,9,5)(2,7,6)(3,8,4), (2,7,3,4)(5,8,9,6), ( 1, 2,10)( 4, 9, 5)( 6, 8, 7), (2,6,4,9,3,8,7,5), (4,7)(5,8)(6,9) ] ), Subgroup( pyl29, [ (1,9,5)(2,7,6)(3,8,4), (2,7,3,4)(5,8,9,6), ( 1, 2,10)( 4, 9, 5)( 6, 8, 7), (2,6,4,9,3,8,7,5) ] ), Subgroup( pyl29, [ (1,9,5)(2,7,6)(3,8,4), (2,7,3,4)(5,8,9,6), ( 1, 2,10)( 4, 9, 5)( 6, 8, 7) ] ), Subgroup( pyl29, [ ] ) ] gap> List( [1..3], i->cs[i].operations.FactorGroup(cs[i],cs[i+1]) ); [ Group( (1,2) ), Group( (1,2) ), Group( (1,9,5)(2,7,6)(3,8,4), (2,7,3,4)(5,8,9,6), ( 1, 2,10) ( 4, 9, 5)( 6, 8, 7) ) ] gap> List( last, Size ); [ 2, 2, 360 ]

`ExponentsPermSolvablePermGroup( `

`G`, `perm` [, `start` ] )

`ExponentsPermSolvablePermGroup`

returns a list `e`, such that

, where `perm` =
`G`.bssgs[1]^`e`[1] {*} `G`.bssgs[2]^`e`[2] {*} ... {*}
`G`.bssgs[`n`]^`e`[`n`]

must be a prime-step
base-strong subnormal generating system as calculated by
`G`.bssgs`ElementaryAbelianSeries`

(see ElementaryAbelianSeries and above). If
the optional third argument `start` is given, the list entries

are left unbound and the element
`exps`[1], ..., `exps`[`start`-1]`perm` is decomposed as product of the remaining pag generators

.
`G`.bssgs[`start`], ..., `G`.bssgs[`n`]

gap> S := Group( (1,2,3,4), (1,2) );; S.name := "s4";; gap> ElementaryAbelianSeries( S );; gap> S.bssgs; [ (1,2), (1,3,2), (1,4)(2,3), (1,2)(3,4) ] gap> ExponentsPermSolvablePermGroup( S, (1,2,3) ); [ 0, 2, 0, 0 ]

This function converts a solvable permutation group into an ag group. It
calculates an elementary abelian series and a prime-step bssgs for `G`
(see `ElementaryAbelianSeries`

above) and then finds the relators
belonging to this prime-step bssgs using the function
`ExponentsPermSolvablePermGroup`

(see above). An isomorphism from the ag
group to the permutation group is bound to `AgGroup( `

.
`G` ).bijection

gap> G := WreathProduct( SymmetricGroup( 4 ), CyclicGroup( 3 ) );; gap> A := AgGroup( G ); Group( g1, g2, g3, g4, g5, g6, g7, g8, g9, g10, g11, g12, g13 ) gap> (A.1*A.3)^A.bijection; ( 1, 6,10, 2, 5, 9)( 3, 7,11)( 4, 8,12)

If `G` and `H` are both permutation groups, `DirectProduct`

constructs the
direct product of `G` and `H` as an intransitive permutation group. There are
special routines for `Centre`

, `Centralizer`

and `SylowSubgroup`

for such
groups that will work faster than the standard permutation group functions.
These functions are `DirectProductPermGroupCentre`

,
`DirectProductPermGroupCentralizer`

and `DirectProductPermGroupSylowSubgroup`

.
You can enforce that these routines will be always used for direct products
of permutation groups by issuing the following three commands (They are not
performed by standard as the code has not been well-tested).

gap> DirectProductPermGroupOps.Centre:=DirectProductPermGroupCentre;; gap> DirectProductPermGroupOps.Centralizer:= > DirectProductPermGroupCentralizer;; gap> DirectProductPermGroupOps.SylowSubgroup:= > DirectProductPermGroupSylowSubgroup;;

All functions that deal with operations of groups are applicable to permutation groups (see Operations of Groups). This section describes which functions are implemented specially for permutation groups. Functions not mentioned here are handled by the default methods described in the respective sections.

`IsSemiRegular`

returns `true`

if `G` operates semiregularly on the
domain `D` and `false`

otherwise.

If *D* is a list of integers and `opr` is `OnPoints`

, `IsSemiRegular`

uses the lemma that says that such an operation is semiregular if all
orbits of *G* on *D* have the same length, and if for an arbitrary point
*p* of *D* and for each generator *g* of *G* there is a permutation *z _{g}*
(not necessarily in

`RepresentativeOperation( `

`G`, `d`, `e`, `opr` )

`RepresentativeOperation`

returns a permutation `perm` in `G` that maps
`d` to `e` in respect to the given operation `opr` if such a permutation
exists, and `false`

otherwise.

If the operation is `OnPoints`

, `OnPairs`

, `OnTuples`

, or `OnSets`

and
`d` and `e` are positive integers or lists of integers, a basechange is
performed and the representative is computed from the factorized inverse
transversal (see Stabilizer Chains and StabChain).

If the operation is `OnPoints`

, `OnPairs`

, `OnTuples`

or `OnSets`

and `d`
and `e` are permutations or lists of permutations, a backtrack search is
performed (see PermGroupOps.ElementProperty).

`Stabilizer`

returns the stabilizer of `D` in `G` using the operation
`opr` on the `D`. If `D` is a positive integer (respectively a list of
positive integers) and the operation `opr` is `OnPoints`

(respectively
`OnPairs`

or `OnTuples`

) a basechange of `G` is performed (see
MakeStabChain). If `D` is a set of positive integers and the operation
`opr` is `OnSets`

a backtrack algorithm for set-stabilizers of
permutation groups is performed.

`Blocks( `

`G`, `D` [, `seed` ] [, `operation` ] )

Returns a partition of `D` being a minimal block system of `G` in respect
to the operation `opreration` on the objects of `D`. If the argument
`seed` is given the objects of `seed` are contained in the same block.
If `D` is a list of positive integers an Atkinson algorithm is performed.

Theoretically the algorithm lies in *{O}(n ^{3} m)* but in practice it
is mostly in

This section describes the various homomorphisms that are treated specially for permutation groups.

`GroupHomomorphisByImages( `

`P`, `H`, `gens`, `imgs` )

The group homomorphism of a permutation group `P` into another group `H`
is handled especially by `GroupHomomorphisByImages`

. Below we describe
how the various mapping functions are implemented for such a group
homomorphism `ghom`. The mapping functions not mentioned below are
implemented by the default functions described in
GroupHomomorphismByImages.

To work with `ghom`, a stabilizer chain for the source of `ghom` is
computed and stored as

, `ghom`.orbit

,
`ghom`.transversal

. For every stabilizer `ghom`.stabilizer`stab` in the stabilizer chain
there is a list parallel to

, which is called
`stab`.generators

, and contains images of the generators. The
stabilizer chain is computed with a random Schreier Sims algorithm, using
the size of the source to know when to stop.
`stab`.genimages

To test if `ghom` is a (single valued) mapping, all Schreier generatores
are computed. Each Schreier generator is then reduced along the
stabilizer chain. Because the chain is complete, each one must reduce to
the identity. Parallel the images of the strong generators are
multiplied. If they also reduce to the identity (in the range), `ghom`
is a function, otherwise the remainders form a normal generating set for
the subgroup of images of the identity of the source.

The image of an element `elm` can be computed by reducing the element
along the stabilizer chain, and at each step multiplying the
corresponding images of the strong generators.

`CompositionMapping( `

`hom`, `ghom` )

The composition of an arbitrary group homomorphism `hom` and `ghom` the
stabilizer chain of `ghom` is copied. On each level the images of the
generators in

are replaced by their images under
`stab`.genimages`hom`.

`OperationHomomorphism( `

`P`, Operation( `P`, `list` ) )

The operation of a permutation group `P` on a list `list` of integers is
handled especially by `OperationHomomorphism`

. (Note that `list` must be
a union of orbits of `P` for `Operation`

to work.) We call the resulting
homomorphism a **transitive constituent** homomorphism. Below we describe
how the various mapping functions are implemented for a transitive
constituent homomorphism `tchom`. The mapping functions not mentioned
below are implemented by the default functions described in
OperationHomomorphism.

The image of an element is computed by restricting `elm` to `list` (see
RestrictedPerm) and conjugating the restricted permutation with

, which maps it to a permutation that operates on
`tchom`.conperm`[1..Length(`

instead of `list`)]`list`.

`Image( `

`tchom`, `H` )

The image of a subgroup `H` is computed as follows. First a stabilizer
chain for `H` is computed. This stabilizer chain is such that the base
starts with points in `list`. Then the images of the strong generators
of `sub` form a strong generating set of the image.

The preimage of a subgroup `H` is computed as follows. First a
stabilizer chain for the source of `tchom` is computed. This stabilizer
chain is such that the base starts with the point in `list`. Then the
kernel of `tchom` is a stabilizer in this stabilizer chain. The
preimages of the strong generators for `H` together with the strong
generators for the kernel form a strong generating set of the preimage
subgroup.

`OperationHomomorphism( `

`P`, Operation( `P`, `blocks`, OnSets ) )

The operation of a permutation group `P` on a block system `blocks` (see
Blocks) is handled especially by `OperationHomomorphism`

. We call the
resulting homomorphism a **blocks homomorphism**. Below we describe how
the various mapping functions are implemented for a blocks homomorphism
`bhom`. The mapping functions not mentioned below are implemented by the
default functions described in OperationHomomorphism.

To compute the image of an element `elm` under `bhom`, the record for
`bhom` contains a list

, which contains for each point in
the union of the blocks the position of this block in `bhom`.reps`blocks`. Then the
image of an element can simply be computed by applying the element to a
representative of each block and using

to find in which
block the image lies.
`bhom`.reps

`Image( `

`bhom`, `H` )

`PreImage( `

`bhom`, `elm` )

`PreImage( `

`bhom`, `H` )

`Kernel( `

`bhom` )

The image of a subgroup, the preimage of an element, and the preimage of a subgroup are computed by rather complicated algorithms. For a description of these algorithms see But85a.

When permutation groups become larger, computations become slower. This increase might make it impossible to compute with these groups. The reason is mainly the creation of stabilizer chains (see StabChain): During this process a lot of schreier generators are produced for the next point stabilizer in the chain, and these generators must be processed. In actual examples, it is observed, however, that much fewer generators are needed. This observation can be justified theoretically and the random methods exploit it by using a method of the Schreier-Sims algorithm which gives the correct result with an user-given error probability.

**Advantage:**:

Computations become much faster. In fact, large problems may be handled only by using random methods.

**Disadvantages:**:

Computations might produce wrong results. However, you can set an error margin, which is guaranteed. The practical performance is even better than our guarantee. You should also keep in mind, that it is impossible, to eliminate system, user or programming errors.

However, there are many situations, when theory offers methods to check
correctness of the results. As an example, consider the following situation.
You want to compute some maximal subgroups of large sporadic groups. The
ATLAS of finite groups then tells you the sizes of the groups as well as the
sizes of the subgroups. The error of the random methods is one-sided in the
sense that they never create strong generators which are not elements of the
group. Hence if the resulting group
sizes are correct, you have indeed obtained the correct result.
You might also give this information to `StabChain`

, and computation will
not only be much faster, but also corresponding to the information, i.e. if
you give the size, the stabilizer chain is computed correctly.

The stabilizer chain is computed using methods from BCFS91.

**How to use the random methods**

**GAP3** provides the global variable `StabChainOptions`

. This record might
contain a component `random`

. If it is set to a number *i* between 1 and
1000 at the beginning, random methods with guaranteed correctness
*i/10* percent are used (though practically the probability for
correctness is much higher). This means that at all applicable
places random methods will be used automatically by the same function calls.
If the component is not set or set to 1000, all computations are
deterministic. By standard, this component is not set, so unless you
explicitely allow random computations none are used.

If the group acts on not more than a hundered points, the use of random methods has no advantage. For these groups always the deterministic methods are used.

gap> g:=SL(4,7); SL(4,7) gap> o:=Orbit(g,[1,0,0,0]*Z(7)^0,OnLines);;Length(o); 400 gap> op:=Operation(g,o,OnLines);;

We create a large permutation group on 400 points. First we compute deterministic.

gap> g:=Group(op.generators,());; gap> StabChain(g);;time; 164736 gap> Size(g); 2317591180800

Now random methods will be used. We allow that the result is guaranteed correct only with 10 percent probability. The group is created anew.

gap> StabChainOptions.random:=100; 100 gap> g:=Group(op.generators,());; gap> StabChain(g);;time; 10350 gap> Size(g); 2317591180800

The result is still correct, though it took only less than one tenth of the time (your mileage may vary). If you give the algorithm a chance to check its results, things become even faster.

gap> g:=Group(op.generators,());; gap> StabChain(g,rec(size:=2317591180800));;time; 5054

**More about random methods**

When stabilizer chains are created, while random methods are allowed, it is
noted in the respective groups, by setting of a record component
`G``.stabChainOptions`

, which is itself a record, containg the component
`random`

. This component has the value indicated by `StabChainOptions`

at the
time the group was created. Values set in this component override
the global setting of `StabChainOptions`

. Whenever stabilizer chains are
created for a group not posessing the `.stabChainOptions.random`

entry, it is
created anew from the global value `StabChainOptions`

.

If a subgroup has no own record `stabChainOptions`

, the one of the parent
group is used instead.

As errors induced by the random functions might propagate, any
(applicable) object created from the group inherits the component
`.stabChainOptions`

from the group. This applies for example to
`Operation`

s and Homomorphisms.

All groups are represented by a record that contains information about the group. A permutation group record contains the following components in addition to those described in section Group Records.

`isPermGroup`

:-

always`true`

.

`isFinite`

:-

always`true`

as permutation groups are always of finite order.

A stabilizer chain (see Stabilizer Chains) is stored recursively in
**GAP3**. The group record of a permutation group `G` with a stabilizer
chain has the following additional components.

`orbit`

:-

List of orbitpoints of`orbit[1]`

(which is the basepoint) under the action of the generators.

`transversal`

:-

Factorized inverse transversal as defined in Stabilizer Chains.

`stabilizer`

:-

Record for the stabilizer of the point`orbit[1]`

in the group generated by`generators`

. The components of this record are again`generators`

,`orbit`

,`transversal`

and`stabilizer`

. The last stabilizer in the stabilizer chain only contains the component`generators`

, which is an empty list.

`stabChainOptions`

:-

A record, that contains information about creation of the stabilizer chain. For example, whether it has been computed using random methods (see Random Methods for Permutation Groups). Some functions also use this record for passing local information about basechanges.

`stabChain`

:-

A record, that contains all information about the stabilizer chain. Functions acessing the stabilizer chain should do it using this record, as it is planned to remove the above three components from the group record in the future. The components of the`stabChain`

record are described below.

The components of the `stabChain`

record for a group `G` are

`identity`

:-

Contains`G``.identity`

.

`generators`

:-

Contains a copy of the generators of`G`, created by`ShallowCopy(G.generators)`

.

`orbit`

:-

is the same as`G``.orbit`

.

`transversal`

:-

is the same as`G``.transversal`

.

`stabilizer`

:-

is the same as`G``.stabilizer`

.

Note that the values of all these components are changed by functions that change, extend, or reduce a base (see MakeStabChain, ExtendStabChain, and ReduceStabChain).

Note that the records that represent the stabilizers are not themselves
group records (see Group Records). Thus you cannot take such a
stabilizer and apply group functions to it. The last `stabilizer`

in the
stabilizer chain is a record whose component `generators`

is empty.

gap3-jm

02 Dec 2021