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41 Group Actions

41.12 External Sets

41.12-1 IsExternalSet

41.12-2 ExternalSet

41.12-3 ActingDomain

41.12-4 FunctionAction

41.12-5 HomeEnumerator

41.12-6 IsExternalSubset

41.12-7 ExternalSubset

41.12-8 IsExternalOrbit

41.12-9 ExternalOrbit

41.12-10 StabilizerOfExternalSet

41.12-11 ExternalOrbits

41.12-12 ExternalOrbitsStabilizers

41.12-13 CanonicalRepresentativeOfExternalSet

41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet

41.12-15 ActorOfExternalSet

41.12-16 UnderlyingExternalSet

41.12-17 SurjectiveActionHomomorphismAttr

41.12-1 IsExternalSet

41.12-2 ExternalSet

41.12-3 ActingDomain

41.12-4 FunctionAction

41.12-5 HomeEnumerator

41.12-6 IsExternalSubset

41.12-7 ExternalSubset

41.12-8 IsExternalOrbit

41.12-9 ExternalOrbit

41.12-10 StabilizerOfExternalSet

41.12-11 ExternalOrbits

41.12-12 ExternalOrbitsStabilizers

41.12-13 CanonicalRepresentativeOfExternalSet

41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet

41.12-15 ActorOfExternalSet

41.12-16 UnderlyingExternalSet

41.12-17 SurjectiveActionHomomorphismAttr

A *group action* is a triple (G, Ω, μ), where G is a group, Ω a set and μ : Ω × G → Ω a function that is compatible with the group arithmetic. We call Ω the *domain* of the action.

In **GAP**, Ω can be a duplicate-free collection (an object that permits access to its elements via the Ω[n] operation, for example a list), it does not need to be sorted (see `IsSet`

(21.17-4)).

The acting function μ is a binary **GAP** function that returns the image μ( x, g ) for a point x ∈ Ω and a group element g ∈ G.

In **GAP**, groups always act from the right, that is μ( μ( x, g ), h ) = μ( x, gh ).

**GAP** does not test whether the acting function μ satisfies the conditions for a group operation but silently assumes that is does. (If it does not, results are unpredictable.)

The first section of this chapter, 41.1, describes the various ways how operations for group actions can be called.

Functions for several commonly used action are already built into **GAP**. These are listed in section 41.2.

The sections 41.7 and 41.8 describe homomorphisms and mappings associated to group actions as well as the permutation group image of an action.

The other sections then describe operations to compute orbits, stabilizers, as well as properties of actions.

Finally section 41.12 describes the concept of "external sets" which represent the concept of a *G-set* and underly the actions mechanism.

The syntax which is used by the operations for group actions is quite flexible. For example we can call the operation `OrbitsDomain`

(41.4-3) for the orbits of the group `G` on the domain `Omega` in the following ways:

`OrbitsDomain`

( G, Ω[, μ] )The acting function μ is optional. If it is not given, the built-in action

`OnPoints`

(41.2-1) (which defines an action via the caret operator`^`

) is used as a default.`OrbitsDomain`

( G, Ω, gens, acts[, μ] )This second version of

`OrbitsDomain`

(41.4-3) permits one to implement an action induced by a homomorphism: If the group H acts on Ω via μ and φ : G → H is a homomorphism, G acts on Ω via the induced action μ'( x, g ) = μ( x, g^φ ).Here gens must be a set of generators of G and acts the images of gens under φ. μ is the acting function for H. Again, the function μ is optional and

`OnPoints`

(41.2-1) is used as a default.The advantage of this notation is that

**GAP**does not need to construct this homomorphism φ and the range group`H`as**GAP**objects. (If a small group G acts via complicated objects acts this otherwise could lead to performance problems.)**GAP**does not test whether the mapping gens ↦ acts actually induces a homomorphism and the results are unpredictable if this is not the case.`OrbitsDomain`

( extset )A third variant is to call the operation with an external set, which then provides G, Ω and μ. You will find more about external sets in Section 41.12.

For operations like `Stabilizer`

(41.5-2) of course the domain must be replaced by an element of the domain of the action.

**GAP** already provides acting functions for the more common actions of a group. For built-in operations such as `Stabilizer`

(41.5-2) special methods are available for many of these actions.

If one needs an action for which no acting function is provided by the library it can be implemented via a **GAP** function that conforms to the syntax

`actfun( omega, g )`

where `omega`

is an element of the action domain, `g`

is an element of the acting group, and the return value is the image of `omega`

under `g`

.

For example one could define the following function that acts on pairs of polynomials via `OnIndeterminates`

(41.2-13):

OnIndeterminatesPairs:= function( polypair, g ) return [ OnIndeterminates( polypair[1], g ), OnIndeterminates( polypair[2], g ) ]; end;

Note that this function *must* implement a group action from the *right*. This is not verified by **GAP** and results are unpredictable otherwise.

`‣ OnPoints` ( pnt, g ) | ( function ) |

returns

. This is for example the action of a permutation group on points, or the action of a group on its elements via conjugation. The action of a matrix group on vectors from the right is described by both `pnt` ^ `g``OnPoints`

and `OnRight`

(41.2-2).

`‣ OnRight` ( pnt, g ) | ( function ) |

returns

. This is for example the action of a group on its elements via right multiplication, or the action of a group on the cosets of a subgroup. The action of a matrix group on vectors from the right is described by both `pnt` * `g``OnPoints`

(41.2-1) and `OnRight`

.

`‣ OnLeftInverse` ( pnt, g ) | ( function ) |

returns `g`^{-1} `* `

. Forming the inverse is necessary to make this a proper action, as in `pnt`**GAP** groups always act from the right.

`OnLeftInverse`

is used for example in the representation of a right coset as an external set (see 41.12), that is, a right coset Ug is an external set for the group U acting on it via `OnLeftInverse`

.)

`‣ OnSets` ( set, g ) | ( function ) |

Let `set` be a proper set (see 21.19). `OnSets`

returns the proper set formed by the images of all points x of `set` via the action function `OnPoints`

(41.2-1), applied to x and `g`.

`OnSets`

is for example used to compute the action of a permutation group on blocks.

(`OnTuples`

(41.2-5) is an action on lists that preserves the ordering of entries.)

`‣ OnTuples` ( tup, g ) | ( function ) |

Let `tup` be a list. `OnTuples`

returns the list formed by the images of all points x of `tup` via the action function `OnPoints`

(41.2-1), applied to x and `g`.

(`OnSets`

(41.2-4) is an action on lists that additionally sorts the entries of the result.)

`‣ OnPairs` ( tup, g ) | ( function ) |

is a special case of `OnTuples`

(41.2-5) for lists `tup` of length 2.

`‣ OnSetsSets` ( set, g ) | ( function ) |

implements the action on sets of sets. For the special case that the sets are pairwise disjoint, it is possible to use `OnSetsDisjointSets`

(41.2-8). `set` must be a sorted list whose entries are again sorted lists, otherwise an error is triggered (see 41.3).

`‣ OnSetsDisjointSets` ( set, g ) | ( function ) |

implements the action on sets of pairwise disjoint sets (see also `OnSetsSets`

(41.2-7)). `set` must be a sorted list whose entries are again sorted lists, otherwise an error is triggered (see 41.3).

`‣ OnSetsTuples` ( set, g ) | ( function ) |

implements the action on sets of tuples. `set` must be a sorted list, otherwise an error is triggered (see 41.3).

`‣ OnTuplesSets` ( set, g ) | ( function ) |

implements the action on tuples of sets. `set` must be a list whose entries are again sorted lists, otherwise an error is triggered (see 41.3).

`‣ OnTuplesTuples` ( set, g ) | ( function ) |

implements the action on tuples of tuples.

gap> g:=Group((1,2,3),(2,3,4));; gap> Orbit(g,1,OnPoints); [ 1, 2, 3, 4 ] gap> Orbit(g,(),OnRight); [ (), (1,2,3), (2,3,4), (1,3,2), (1,3)(2,4), (1,2)(3,4), (2,4,3), (1,4,2), (1,4,3), (1,3,4), (1,2,4), (1,4)(2,3) ] gap> Orbit(g,[1,2],OnPairs); [ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ 3, 4 ], [ 2, 1 ], [ 1, 4 ], [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ] gap> Orbit(g,[1,2],OnSets); [ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ] gap> Orbit(g,[[1,2],[3,4]],OnSetsSets); [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 1, 3 ], [ 2, 4 ] ] ] gap> Orbit(g,[[1,2],[3,4]],OnTuplesSets); [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 2, 4 ] ], [ [ 3, 4 ], [ 1, 2 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 4 ], [ 1, 3 ] ] ] gap> Orbit(g,[[1,2],[3,4]],OnSetsTuples); [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 1, 3 ], [ 4, 2 ] ], [ [ 2, 4 ], [ 3, 1 ] ], [ [ 2, 1 ], [ 4, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ] ] gap> Orbit(g,[[1,2],[3,4]],OnTuplesTuples); [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 4, 2 ] ], [ [ 3, 1 ], [ 2, 4 ] ], [ [ 3, 4 ], [ 1, 2 ] ], [ [ 2, 1 ], [ 4, 3 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 2 ] ], [ [ 4, 2 ], [ 1, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ], [ [ 2, 4 ], [ 3, 1 ] ], [ [ 4, 3 ], [ 2, 1 ] ] ]

`‣ OnLines` ( vec, g ) | ( function ) |

Let `vec` be a *normed* row vector, that is, its first nonzero entry is normed to the identity of the relevant field, see `NormedRowVector`

(23.2-1). The function `OnLines`

returns the row vector obtained from first multiplying `vec` from the right with `g` (via `OnRight`

(41.2-2)) and then normalizing the resulting row vector by scalar multiplication from the left.

This action corresponds to the projective action of a matrix group on one-dimensional subspaces.

If `vec` is a zero vector or is not normed then an error is triggered (see 41.3).

gap> gl:=GL(2,5);;v:=[1,0]*Z(5)^0; [ Z(5)^0, 0*Z(5) ] gap> h:=Action(gl,Orbit(gl,v,OnLines),OnLines); Group([ (2,3,5,6), (1,2,4)(3,6,5) ])

`‣ OnIndeterminates` ( poly, perm ) | ( function ) |

A permutation `perm` acts on the multivariate polynomial `poly` by permuting the indeterminates as it permutes points.

gap> x:=Indeterminate(Rationals,1);; y:=Indeterminate(Rationals,2);; gap> OnIndeterminates(x^7*y+x*y^4,(1,17)(2,28)); x_17^7*x_28+x_17*x_28^4 gap> Stabilizer(Group((1,2,3,4),(1,2)),x*y,OnIndeterminates); Group([ (1,2), (3,4) ])

`‣ Permuted` ( list, perm ) | ( method ) |

The following example demonstrates `Permuted`

(21.20-18) being used to implement a permutation action on a domain:

gap> g:=Group((1,2,3),(1,2));; gap> dom:=[ "a", "b", "c" ];; gap> Orbit(g,dom,Permuted); [ [ "a", "b", "c" ], [ "c", "a", "b" ], [ "b", "a", "c" ], [ "b", "c", "a" ], [ "a", "c", "b" ], [ "c", "b", "a" ] ]

`‣ OnSubspacesByCanonicalBasis` ( bas, mat ) | ( function ) |

`‣ OnSubspacesByCanonicalBasisConcatenations` ( basvec, mat ) | ( function ) |

implements the operation of a matrix group on subspaces of a vector space. `bas` must be a list of (linearly independent) vectors which forms a basis of the subspace in Hermite normal form. `mat` is an element of the acting matrix group. The function returns a mutable matrix which gives the basis of the image of the subspace in Hermite normal form. (In other words: it triangulizes the product of `bas` with `mat`.)

`bas` must be given in Hermite normal form, otherwise an error is triggered (see 41.3).

A variety of action functions assumes that the objects on which it acts are given in a particular form, for example canonical representatives. Affected actions are for example `OnSetsSets`

(41.2-7), `OnSetsDisjointSets`

(41.2-8), `OnSetsTuples`

(41.2-9), `OnTuplesSets`

(41.2-10), `OnLines`

(41.2-12) and `OnSubspacesByCanonicalBasis`

(41.2-15).

If orbit seeds or domain elements are not given in the required form **GAP** will issue an error message:

gap> Orbit(SymmetricGroup(5),[[2,4],[1,3]],OnSetsSets); Error, Action not well-defined. See the manual section ``Action on canonical representatives''.

In this case the affected domain elements have to be brought in canonical form, as documented for the respective action function. For interactive use this is most easily done by acting with the identity element of the group.

(A similar error could arise if a user-defined action function is used which actually does not implement an action from the right.)

If a group G acts on a set Ω, the set of all images of x ∈ Ω under elements of G is called the *orbit* of x. The set of orbits of G is a partition of Ω.

`‣ Orbit` ( G[, Omega], pnt[, gens, acts][, act] ) | ( operation ) |

The orbit of the point `pnt` is the list of all images of `pnt` under the action of the group `G` w.r.t. the action function `act` or `OnPoints`

(41.2-1) if no action function is given.

(Note that the arrangement of points in this list is not defined by the operation.)

The orbit of `pnt` will always contain one element that is *equal* to `pnt`, however for performance reasons this element is not necessarily *identical* to `pnt`, in particular if `pnt` is mutable.

gap> g:=Group((1,3,2),(2,4,3));; gap> Orbit(g,1); [ 1, 3, 2, 4 ] gap> Orbit(g,[1,2],OnSets); [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ], [ 2, 4 ] ]

(See Section 41.2 for information about specific actions.)

`‣ Orbits` ( G, seeds[, gens, acts][, act] ) | ( operation ) |

`‣ Orbits` ( xset ) | ( attribute ) |

returns a duplicate-free list of the orbits of the elements in `seeds` under the action `act` of `G` or under `OnPoints`

(41.2-1) if no action function is given.

(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)

`‣ OrbitsDomain` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ OrbitsDomain` ( xset ) | ( attribute ) |

returns a list of the orbits of `G` on the domain `Omega` (given as lists) under the action `act` or under `OnPoints`

(41.2-1) if no action function is given.

This operation is often faster than `Orbits`

(41.4-2). The domain `Omega` must be closed under the action of `G`, otherwise an error can occur.

(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)

gap> g:=Group((1,3,2),(2,4,3));; gap> Orbits(g,[1..5]); [ [ 1, 3, 2, 4 ], [ 5 ] ] gap> OrbitsDomain(g,Arrangements([1..4],3),OnTuples); [ [ [ 1, 2, 3 ], [ 3, 1, 2 ], [ 1, 4, 2 ], [ 2, 3, 1 ], [ 2, 1, 4 ], [ 3, 4, 1 ], [ 1, 3, 4 ], [ 4, 2, 1 ], [ 4, 1, 3 ], [ 2, 4, 3 ], [ 3, 2, 4 ], [ 4, 3, 2 ] ], [ [ 1, 2, 4 ], [ 3, 1, 4 ], [ 1, 4, 3 ], [ 2, 3, 4 ], [ 2, 1, 3 ], [ 3, 4, 2 ], [ 1, 3, 2 ], [ 4, 2, 3 ], [ 4, 1, 2 ], [ 2, 4, 1 ], [ 3, 2, 1 ], [ 4, 3, 1 ] ] ] gap> OrbitsDomain(g,GF(2)^2,[(1,2,3),(1,4)(2,3)], > [[[Z(2)^0,Z(2)^0],[Z(2)^0,0*Z(2)]],[[Z(2)^0,0*Z(2)],[0*Z(2),Z(2)^0]]]); [ [ <an immutable GF2 vector of length 2> ], [ <an immutable GF2 vector of length 2>, <an immutable GF2 vector of length 2>, <an immutable GF2 vector of length 2> ] ]

(See Section 41.2 for information about specific actions.)

`‣ OrbitLength` ( G, Omega, pnt[, gens, acts][, act] ) | ( operation ) |

computes the length of the orbit of `pnt` under the action function `act` or `OnPoints`

(41.2-1) if no action function is given.

`‣ OrbitLengths` ( G, seeds[, gens, acts][, act] ) | ( operation ) |

`‣ OrbitLengths` ( xset ) | ( attribute ) |

computes the lengths of all the orbits of the elements in `seeds` under the action `act` of `G`.

`‣ OrbitLengthsDomain` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ OrbitLengthsDomain` ( xset ) | ( attribute ) |

computes the lengths of all the orbits of `G` on `Omega`.

This operation is often faster than `OrbitLengths`

(41.4-5). The domain `Omega` must be closed under the action of `G`, otherwise an error can occur.

gap> g:=Group((1,3,2),(2,4,3));; gap> OrbitLength(g,[1,2,3,4],OnTuples); 12 gap> OrbitLengths(g,Arrangements([1..4],4),OnTuples); [ 12, 12 ]

The *stabilizer* of a point x under the action of a group G is the set of all those elements in G which fix x.

`‣ OrbitStabilizer` ( G[, Omega], pnt[, gens, acts], act ) | ( operation ) |

computes the orbit and the stabilizer of `pnt` simultaneously in a single orbit-stabilizer algorithm.

The stabilizer will have `G` as its parent.

`‣ Stabilizer` ( G[, Omega], pnt[, gens, acts][, act] ) | ( function ) |

computes the stabilizer in `G` of the point `pnt`, that is the subgroup of those elements of `G` that fix `pnt`. The stabilizer will have `G` as its parent.

gap> g:=Group((1,3,2),(2,4,3));; gap> Stabilizer(g,4); Group([ (1,3,2) ])

The stabilizer of a set or tuple of points can be computed by specifying an action of sets or tuples of points.

gap> Stabilizer(g,[1,2],OnSets); Group([ (1,2)(3,4) ]) gap> Stabilizer(g,[1,2],OnTuples); Group(()) gap> OrbitStabilizer(g,[1,2],OnSets); rec( orbit := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ], [ 2, 4 ] ], stabilizer := Group([ (1,2)(3,4) ]) )

(See Section 41.2 for information about specific actions.)

The standard methods for all these actions are an orbit-stabilizer algorithm. For permutation groups backtrack algorithms are used. For solvable groups an orbit-stabilizer algorithm for solvable groups, which uses the fact that the orbits of a normal subgroup form a block system (see [LNS84]) is used.

`‣ OrbitStabilizerAlgorithm` ( G, Omega, blist, gens, acts, pntact ) | ( operation ) |

This operation should not be called by a user. It is documented however for purposes to extend or maintain the group actions package (the word "package" here refers to the **GAP** functionality for group actions, not to a **GAP** package).

`OrbitStabilizerAlgorithm`

performs an orbit stabilizer algorithm for the group `G` acting with the generators `gens` via the generator images `gens` and the group action `act` on the element `pnt`. (For technical reasons `pnt` and `act` are put in one record with components `pnt`

and `act`

respectively.)

The `pntact` record may carry a component `stabsub`. If given, this must be a subgroup stabilizing *all* points in the domain and can be used to abbreviate stabilizer calculations.

The `pntact` component also may contain the boolean entry `onlystab`

set to `true`

. In this case the `orbit`

component may be omitted from the result.

The argument `Omega` (which may be replaced by `false`

to be ignored) is the set within which the orbit is computed (once the orbit is the full domain, the orbit calculation may stop). If `blist` is given it must be a bit list corresponding to `Omega` in which elements which have been found already will be "ticked off" with `true`

. (In particular, the entries for the orbit of `pnt` still must be all set to `false`

). Again the remaining action domain (the bits set initially to `false`

) can be used to stop if the orbit cannot grow any longer. Another use of the bit list is if `Omega` is an enumerator which can determine `PositionCanonical`

(21.16-3) values very quickly. In this situation it can be worth to search images not in the orbit found so far, but via their position in `Omega` and use a the bit list to keep track whether the element is in the orbit found so far.

`‣ RepresentativeAction` ( G[, Omega], d, e[, gens, acts][, act] ) | ( function ) |

computes an element of `G` that maps `d` to `e` under the given action and returns `fail`

if no such element exists.

gap> g:=Group((1,3,2),(2,4,3));; gap> RepresentativeAction(g,1,3); (1,3)(2,4) gap> RepresentativeAction(g,1,3,OnPoints); (1,3)(2,4) gap> RepresentativeAction(g,(1,2,3),(2,4,3)); (1,2,4) gap> RepresentativeAction(g,(1,2,3),(2,3,4)); fail gap> RepresentativeAction(g,Group((1,2,3)),Group((2,3,4))); (1,2,4) gap> RepresentativeAction(g,[1,2,3],[1,2,4],OnSets); (2,4,3) gap> RepresentativeAction(g,[1,2,3],[1,2,4],OnTuples); fail

(See Section 41.2 for information about specific actions.)

Again the standard method for `RepresentativeAction`

is an orbit-stabilizer algorithm, for permutation groups and standard actions a backtrack algorithm is used.

When a group G acts on a domain Ω, an enumeration of Omega yields a homomorphism from G into the symmetric group on { 1, ..., |Ω| }. In **GAP**, the enumeration of Ω is provided by the `Enumerator`

(30.3-2) value of Ω which of course is Ω itself if it is a list.

For an action homomorphism, the operation `UnderlyingExternalSet`

(41.12-16) will return the external set on Ω which affords the action.

`‣ ActionHomomorphism` ( G, Omega[, gens, acts][, act][, "surjective"] ) | ( function ) |

`‣ ActionHomomorphism` ( xset[, "surjective"] ) | ( function ) |

`‣ ActionHomomorphism` ( action ) | ( function ) |

computes a homomorphism from `G` into the symmetric group on |`Omega`| points that gives the permutation action of `G` on `Omega`. (In particular, this homomorphism is a permutation equivalence, that is the permutation image of a group element is given by the positions of points in `Omega`.)

By default the homomorphism returned by `ActionHomomorphism`

is not necessarily surjective (its `Range`

(32.3-7) value is the full symmetric group) to avoid unnecessary computation of the image. If the optional string argument `"surjective"`

is given, a surjective homomorphism is created.

The third version (which is supported only for **GAP**3 compatibility) returns the action homomorphism that belongs to the image obtained via `Action`

(41.7-2).

(See Section 41.2 for information about specific actions.)

gap> g:=Group((1,2,3),(1,2));; gap> hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples); <action homomorphism> gap> Image(hom); Group( [ (1,9,13)(2,10,14)(3,7,15)(4,8,16)(5,12,17)(6,11,18)(19,22,23)(20,21, 24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,15)(14,16)(17,18)(19, 21)(20,22)(23,24) ]) gap> Size(Range(hom));Size(Image(hom)); 620448401733239439360000 6 gap> hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples, > "surjective");; gap> Size(Range(hom)); 6

When acting on a domain, the operation `PositionCanonical`

(21.16-3) is used to determine the position of elements in the domain. This can be used to act on a domain given by a list of representatives for which `PositionCanonical`

(21.16-3) is implemented, for example the return value of `RightTransversal`

(39.8-1).

`‣ Action` ( G, Omega[, gens, acts][, act] ) | ( function ) |

`‣ Action` ( xset ) | ( function ) |

returns the image group of `ActionHomomorphism`

(41.7-1) called with the same parameters.

Note that (for compatibility reasons to be able to get the action homomorphism) this image group internally stores the action homomorphism. If `G` or `Omega` are extremely big, this can cause memory problems. In this case compute only generator images and form the image group yourself.

(See Section 41.2 for information about specific actions.)

The following code shows for example how to create the regular action of a group.

gap> g:=Group((1,2,3),(1,2));; gap> Action(g,AsList(g),OnRight); Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])

`‣ SparseActionHomomorphism` ( G, start[, gens, acts][, act] ) | ( operation ) |

`‣ SortedSparseActionHomomorphism` ( G, start[, gens, acts][, act] ) | ( operation ) |

`SparseActionHomomorphism`

computes the action homomorphism (see `ActionHomomorphism`

(41.7-1)) with arguments `G`, D, and the optional arguments given, where D is the union of the `G`-orbits of all points in `start`. In the `Orbit`

(41.4-1) calls that are used to create D, again the optional arguments given are entered.)

If `G` acts on a very large domain not surjectively this may yield a permutation image of substantially smaller degree than by action on the whole domain.

The operation `SparseActionHomomorphism`

will only use `\=`

(31.11-1) comparisons of points in the orbit. Therefore it can be used even if no good `\<`

(31.11-1) comparison method for these points is available. However the image group will depend on the generators `gens` of `G`.

The operation `SortedSparseActionHomomorphism`

in contrast will sort the orbit and thus produce an image group which does not depend on these generators.

gap> h:=Group(Z(3)*[[[1,1],[0,1]]]); Group([ [ [ Z(3), Z(3) ], [ 0*Z(3), Z(3) ] ] ]) gap> hom:=ActionHomomorphism(h,GF(3)^2,OnRight);; gap> Image(hom); Group([ (2,3)(4,9,6,7,5,8) ]) gap> hom:=SparseActionHomomorphism(h,[Z(3)*[1,0]],OnRight);; gap> Image(hom); Group([ (1,2,3,4,5,6) ])

Of particular importance is the action of a group on its elements or cosets of a subgroup. These actions can be obtained by using `ActionHomomorphism`

(41.7-1) for a suitable domain (for example a list of subgroups). For the following (frequently used) types of actions however special (often particularly efficient) functions are provided. A special case is the regular action on all elements.

`‣ FactorCosetAction` ( G, U[, N] ) | ( operation ) |

This command computes the action of the group `G` on the right cosets of the subgroup `U`. If a normal subgroup `N` of `G` is given, it is stored as kernel of this action.

gap> g:=Group((1,2,3,4,5),(1,2));;u:=SylowSubgroup(g,2);;Index(g,u); 15 gap> FactorCosetAction(g,u); <action epimorphism> gap> StructureDescription(Range(last)); "S5"

`‣ RegularActionHomomorphism` ( G ) | ( attribute ) |

returns an isomorphism from `G` onto the regular permutation representation of `G`.

`‣ AbelianSubfactorAction` ( G, M, N ) | ( operation ) |

Let `G` be a group and `M` ≥ `N` be subgroups of a common parent that are normal under `G`, such that the subfactor `M`/`N` is elementary abelian. The operation `AbelianSubfactorAction`

returns a list `[ `

where `phi`, `alpha`, `bas` ]`bas` is a list of elements of `M` which are representatives for a basis of `M`/`N`, `alpha` is a map from `M` into a n-dimensional row space over GF(p) where [`M`:`N`] = p^n that is the natural homomorphism of `M` by `N` with the quotient represented as an additive group. Finally `phi` is a homomorphism from `G` into GL_n(p) that represents the action of `G` on the factor `M`/`N`.

Note: If only matrices for the action are needed, `LinearActionLayer`

(45.14-3) might be faster.

gap> g:=Group((1,8,10,7,3,5)(2,4,12,9,11,6), > (1,9,5,6,3,10)(2,11,12,8,4,7));; gap> c:=ChiefSeries(g);;List(c,Size); [ 96, 48, 16, 4, 1 ] gap> HasElementaryAbelianFactorGroup(c[3],c[4]); true gap> SetName(c[3],"my_group");; gap> a:=AbelianSubfactorAction(g,c[3],c[4]); [ [ (1,8,10,7,3,5)(2,4,12,9,11,6), (1,9,5,6,3,10)(2,11,12,8,4,7) ] -> [ <an immutable 2x2 matrix over GF2>, <an immutable 2x2 matrix over GF2> ], MappingByFunction( my_group, ( GF(2)^ 2 ), function( e ) ... end, function( r ) ... end ), Pcgs([ (2,9,3,8)(4,11,5,10), (1,6,12,7)(4,10,5,11) ]) ] gap> mat:=Image(a[1],g); Group([ <an immutable 2x2 matrix over GF2>, <an immutable 2x2 matrix over GF2> ]) gap> Size(mat); 3 gap> e:=PreImagesRepresentative(a[2],[Z(2),0*Z(2)]); (2,9,3,8)(4,11,5,10) gap> e in c[3];e in c[4]; true false

If only the permutation image of a single element is needed, it might not be worth to create the action homomorphism, the following operations yield the permutation image and cycles of a single element.

`‣ Permutation` ( g, Omega[, gens, acts][, act] ) | ( function ) |

`‣ Permutation` ( g, xset ) | ( function ) |

computes the permutation that corresponds to the action of `g` on the permutation domain `Omega` (a list of objects that are permuted). If an external set `xset` is given, the permutation domain is the `HomeEnumerator`

(41.12-5) value of this external set (see Section 41.12). Note that the points of the returned permutation refer to the positions in `Omega`, even if `Omega` itself consists of integers.

If `g` does not leave the domain invariant, or does not map the domain injectively then `fail`

is returned.

`‣ PermutationCycle` ( g, Omega, pnt[, act] ) | ( function ) |

computes the permutation that represents the cycle of `pnt` under the action of the element `g`.

gap> Permutation([[Z(3),-Z(3)],[Z(3),0*Z(3)]],AsList(GF(3)^2)); (2,7,6)(3,4,8) gap> Permutation((1,2,3)(4,5)(6,7),[4..7]); (1,2)(3,4) gap> PermutationCycle((1,2,3)(4,5)(6,7),[4..7],4); (1,2)

`‣ Cycle` ( g, Omega, pnt[, act] ) | ( function ) |

returns a list of the points in the cycle of `pnt` under the action of the element `g`.

`‣ CycleLength` ( g, Omega, pnt[, act] ) | ( function ) |

returns the length of the cycle of `pnt` under the action of the element `g`.

`‣ Cycles` ( g, Omega[, act] ) | ( function ) |

returns a list of the cycles (as lists of points) of the action of the element `g`.

`‣ CycleLengths` ( g, Omega[, act] ) | ( operation ) |

returns the lengths of all the cycles under the action of the element `g` on `Omega`.

gap> Cycle((1,2,3)(4,5)(6,7),[4..7],4); [ 4, 5 ] gap> CycleLength((1,2,3)(4,5)(6,7),[4..7],4); 2 gap> Cycles((1,2,3)(4,5)(6,7),[4..7]); [ [ 4, 5 ], [ 6, 7 ] ] gap> CycleLengths((1,2,3)(4,5)(6,7),[4..7]); [ 2, 2 ]

`‣ CycleIndex` ( g, Omega[, act] ) | ( function ) |

`‣ CycleIndex` ( G, Omega[, act] ) | ( function ) |

The *cycle index* of a permutation `g` acting on `Omega` is defined as

z(`g`) = s_1^{c_1} s_2^{c_2} ⋯ s_n^{c_n}

where c_k is the number of k-cycles in the cycle decomposition of `g` and the s_i are indeterminates.

The *cycle index* of a group `G` is defined as

Z(`G`) = ( ∑_{g ∈ `G`} z(g) ) / |`G`| .

The indeterminates used by `CycleIndex`

are the indeterminates 1 to n over the rationals (see `Indeterminate`

(66.1-1)).

gap> g:=TransitiveGroup(6,8); S_4(6c) = 1/2[2^3]S(3) gap> CycleIndex(g); 1/24*x_1^6+1/8*x_1^2*x_2^2+1/4*x_1^2*x_4+1/4*x_2^3+1/3*x_3^2

`‣ IsTransitive` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ IsTransitive` ( G ) | ( property ) |

`‣ IsTransitive` ( xset ) | ( property ) |

returns `true`

if the action implied by the arguments is transitive, or `false`

otherwise.

We say that a group `G` acts *transitively* on a domain D if and only if for every pair of points d, e ∈ D there is an element g in `G` such that d^g = e.

For permutation groups, the syntax `IsTransitive(`

is also permitted and tests whether the group is transitive on the points moved by it, that is the group ⟨ (2,3,4),(2,3) ⟩ is transitive (on 3 points).`G`)

`‣ Transitivity` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ Transitivity` ( xset ) | ( attribute ) |

returns the degree k (a non-negative integer) of transitivity of the action implied by the arguments, i.e. the largest integer k such that the action is k-transitive. If the action is not transitive `0`

is returned.

An action is *k-transitive* if every k-tuple of points can be mapped simultaneously to every other k-tuple.

gap> g:=Group((1,3,2),(2,4,3));; gap> IsTransitive(g,[1..5]); false gap> Transitivity(g,[1..4]); 2

`‣ RankAction` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ RankAction` ( xset ) | ( attribute ) |

returns the rank of a transitive action, i.e. the number of orbits of the point stabilizer.

gap> RankAction(g,Combinations([1..4],2),OnSets); 4

`‣ IsSemiRegular` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ IsSemiRegular` ( xset ) | ( property ) |

returns `true`

if the action implied by the arguments is semiregular, or `false`

otherwise.

An action is *semiregular* is the stabilizer of each point is the identity.

`‣ IsRegular` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ IsRegular` ( xset ) | ( property ) |

returns `true`

if the action implied by the arguments is regular, or `false`

otherwise.

An action is *regular* if it is both semiregular (see `IsSemiRegular`

(41.10-4)) and transitive (see `IsTransitive`

(41.10-1)). In this case every point `pnt` of `Omega` defines a one-to-one correspondence between `G` and `Omega`.

gap> IsSemiRegular(g,Arrangements([1..4],3),OnTuples); true gap> IsRegular(g,Arrangements([1..4],3),OnTuples); false

`‣ Earns` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ Earns` ( xset ) | ( attribute ) |

returns a list of the elementary abelian regular (when acting on `Omega`) normal subgroups of `G`.

At the moment only methods for a primitive group `G` are implemented.

`‣ IsPrimitive` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ IsPrimitive` ( xset ) | ( property ) |

returns `true`

if the action implied by the arguments is primitive, or `false`

otherwise.

An action is *primitive* if it is transitive and the action admits no nontrivial block systems. See 41.11.

gap> IsPrimitive(g,Orbit(g,(1,2)(3,4))); true

A *block system* (system of imprimitivity) for the action of a group G on an action domain Ω is a partition of Ω which –as a partition– remains invariant under the action of G.

`‣ Blocks` ( G, Omega[, seed][, gens, acts][, act] ) | ( operation ) |

`‣ Blocks` ( xset[, seed] ) | ( attribute ) |

computes a block system for the action. If `seed` is not given and the action is imprimitive, a minimal nontrivial block system will be found. If `seed` is given, a block system in which `seed` is the subset of one block is computed. The action must be transitive.

gap> g:=TransitiveGroup(8,3); E(8)=2[x]2[x]2 gap> Blocks(g,[1..8]); [ [ 1, 8 ], [ 2, 3 ], [ 4, 5 ], [ 6, 7 ] ] gap> Blocks(g,[1..8],[1,4]); [ [ 1, 4 ], [ 2, 7 ], [ 3, 6 ], [ 5, 8 ] ]

(See Section 41.2 for information about specific actions.)

`‣ MaximalBlocks` ( G, Omega[, seed][, gens, acts][, act] ) | ( operation ) |

`‣ MaximalBlocks` ( xset[, seed] ) | ( attribute ) |

returns a block system that is maximal (i.e., blocks are maximal with respect to inclusion) for the action of `G` on `Omega`. If `seed` is given, a block system is computed in which `seed` is a subset of one block.

gap> MaximalBlocks(g,[1..8]); [ [ 1, 2, 3, 8 ], [ 4, 5, 6, 7 ] ]

`‣ RepresentativesMinimalBlocks` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ RepresentativesMinimalBlocks` ( xset ) | ( attribute ) |

computes a list of block representatives for all minimal (i.e blocks are minimal with respect to inclusion) nontrivial block systems for the action.

gap> RepresentativesMinimalBlocks(g,[1..8]); [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 1, 8 ] ]

`‣ AllBlocks` ( G ) | ( attribute ) |

computes a list of representatives of all block systems for a permutation group `G` acting transitively on the points moved by the group.

gap> AllBlocks(g); [ [ 1, 8 ], [ 1, 2, 3, 8 ], [ 1, 4, 5, 8 ], [ 1, 6, 7, 8 ], [ 1, 3 ], [ 1, 3, 5, 7 ], [ 1, 3, 4, 6 ], [ 1, 5 ], [ 1, 2, 5, 6 ], [ 1, 2 ], [ 1, 2, 4, 7 ], [ 1, 4 ], [ 1, 7 ], [ 1, 6 ] ]

The stabilizer of a block can be computed via the action `OnSets`

(41.2-4):

gap> Stabilizer(g,[1,8],OnSets); Group([ (1,8)(2,3)(4,5)(6,7) ])

If `bs`

is a partition of the action domain, given as a set of sets, the stabilizer under the action `OnSetsDisjointSets`

(41.2-8) returns the largest subgroup which preserves `bs`

as a block system.

gap> g:=Group((1,2,3,4,5,6,7,8),(1,2));; gap> bs:=[[1,2,3,4],[5,6,7,8]];; gap> Stabilizer(g,bs,OnSetsDisjointSets); Group([ (6,7), (5,6), (5,8), (2,3), (3,4)(5,7), (1,4), (1,5,4,8)(2,6,3,7) ])

When considering group actions, sometimes the concept of a *G-set* is used. This is a set Ω endowed with an action of G. The elements of the G-set are the same as those of Ω, however concepts like equality and equivalence of G-sets do not only consider the underlying domain Ω but the group action as well.

This concept is implemented in **GAP** via *external sets*.

The constituents of an external set are stored in the attributes `ActingDomain`

(41.12-3), `FunctionAction`

(41.12-4) and `HomeEnumerator`

(41.12-5).

Most operations for actions are applicable as an attribute for an external set.

The most prominent external subsets are orbits, see `ExternalOrbit`

(41.12-9).

Many subsets of a group, such as conjugacy classes or cosets (see `ConjugacyClass`

(39.10-1) and `RightCoset`

(39.7-1)) are implemented as external orbits.

External sets also are implicitly underlying action homomorphisms, see `UnderlyingExternalSet`

(41.12-16) and `SurjectiveActionHomomorphismAttr`

(41.12-17).

`‣ IsExternalSet` ( obj ) | ( category ) |

An *external set* specifies a group action μ: Ω × G ↦ Ω of a group G on a domain Ω. The external set knows the group, the domain and the actual acting function. Mathematically, an external set is the set Ω, which is endowed with the action of a group G via the group action μ. For this reason **GAP** treats an external set as a domain whose elements are the elements of Ω. An external set is always a union of orbits. Currently the domain Ω must always be finite. If Ω is not a list, an enumerator for Ω is automatically chosen, see `Enumerator`

(30.3-2).

`‣ ExternalSet` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

creates the external set for the action `act` of `G` on `Omega`. `Omega` can be either a proper set, or a domain which is represented as described in 12.4 and 30, or (to use less memory but with a slower performance) an enumerator (see `Enumerator`

(30.3-2) ) of this domain.

gap> g:=Group((1,2,3),(2,3,4));; gap> e:=ExternalSet(g,[1..4]); <xset:[ 1, 2, 3, 4 ]> gap> e:=ExternalSet(g,g,OnRight); <xset:[ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]> gap> Orbits(e); [ [ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,4,2), (1,2,3), (1,3,4), (2,3,4), (1,3,2), (1,4,3), (1,2,4) ] ]

`‣ ActingDomain` ( xset ) | ( attribute ) |

This attribute returns the group with which the external set `xset` was defined.

`‣ FunctionAction` ( xset ) | ( attribute ) |

is the acting function with which the external set `xset` was defined.

`‣ HomeEnumerator` ( xset ) | ( attribute ) |

returns an enumerator of the action domain with which the external set `xset` was defined. For external subsets, this is in general different from the `Enumerator`

(30.3-2) value of `xset`, which enumerates only the subset.

gap> ActingDomain(e); Group([ (1,2,3), (2,3,4) ]) gap> FunctionAction(e)=OnRight; true gap> HomeEnumerator(e); [ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]

`‣ IsExternalSubset` ( obj ) | ( representation ) |

An external subset is the restriction of an external set to a subset of the domain (which must be invariant under the action). It is again an external set.

`‣ ExternalSubset` ( G, xset, start[, gens, acts], act ) | ( operation ) |

constructs the external subset of `xset` on the union of orbits of the points in `start`.

`‣ IsExternalOrbit` ( obj ) | ( representation ) |

An external orbit is an external subset consisting of one orbit.

`‣ ExternalOrbit` ( G, Omega, pnt[, gens, acts], act ) | ( operation ) |

constructs the external subset on the orbit of `pnt`. The `Representative`

(30.4-7) value of this external set is `pnt`.

gap> e:=ExternalOrbit(g,g,(1,2,3)); (1,2,3)^G

`‣ StabilizerOfExternalSet` ( xset ) | ( attribute ) |

computes the stabilizer of the `Representative`

(30.4-7) value of the external set `xset`. The stabilizer will have the acting group of `xset` as its parent.

gap> Representative(e); (1,2,3) gap> StabilizerOfExternalSet(e); Group([ (1,2,3) ])

`‣ ExternalOrbits` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ ExternalOrbits` ( xset ) | ( attribute ) |

computes a list of external orbits that give the orbits of `G`.

gap> ExternalOrbits(g,AsList(g)); [ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]

`‣ ExternalOrbitsStabilizers` ( G, Omega[, gens, acts][, act] ) | ( operation ) |

`‣ ExternalOrbitsStabilizers` ( xset ) | ( attribute ) |

In addition to `ExternalOrbits`

(41.12-11), this operation also computes the stabilizers of the representatives of the external orbits at the same time. (This can be quicker than computing the `ExternalOrbits`

(41.12-11) value first and the stabilizers afterwards.)

gap> e:=ExternalOrbitsStabilizers(g,AsList(g)); [ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ] gap> HasStabilizerOfExternalSet(e[3]); true gap> StabilizerOfExternalSet(e[3]); Group([ (2,4,3) ])

`‣ CanonicalRepresentativeOfExternalSet` ( xset ) | ( attribute ) |

The canonical representative of an external set `xset` may only depend on the defining attributes `G`, `Omega`, `act` of `xset` and (in the case of external subsets) `Enumerator( `

. It must `xset` )*not* depend, e.g., on the representative of an external orbit. **GAP** does not know methods for arbitrary external sets to compute a canonical representative, see `CanonicalRepresentativeDeterminatorOfExternalSet`

(41.12-14).

`‣ CanonicalRepresentativeDeterminatorOfExternalSet` ( xset ) | ( attribute ) |

returns a function that takes as its arguments the acting group and a point. This function returns a list of length 1 or 3, the first entry being the canonical representative and the other entries (if bound) being the stabilizer of the canonical representative and a conjugating element, respectively. An external set is only guaranteed to be able to compute a canonical representative if it has a `CanonicalRepresentativeDeterminatorOfExternalSet`

.

`‣ ActorOfExternalSet` ( xset ) | ( attribute ) |

returns an element mapping `Representative(`

to `xset`)`CanonicalRepresentativeOfExternalSet(`

under the given action.`xset`)

gap> u:=Subgroup(g,[(1,2,3)]);; gap> e:=RightCoset(u,(1,2)(3,4));; gap> CanonicalRepresentativeOfExternalSet(e); (2,4,3) gap> ActorOfExternalSet(e); (1,3,2) gap> FunctionAction(e)((1,2)(3,4),last); (2,4,3)

`‣ UnderlyingExternalSet` ( acthom ) | ( attribute ) |

The underlying set of an action homomorphism `acthom` is the external set on which it was defined.

gap> g:=Group((1,2,3),(1,2));; gap> hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples);; gap> s:=UnderlyingExternalSet(hom); <xset:[[ 1, 2, 3 ],[ 1, 2, 4 ],[ 1, 3, 2 ],[ 1, 3, 4 ],[ 1, 4, 2 ], [ 1, 4, 3 ],[ 2, 1, 3 ],[ 2, 1, 4 ],[ 2, 3, 1 ],[ 2, 3, 4 ], [ 2, 4, 1 ],[ 2, 4, 3 ],[ 3, 1, 2 ],[ 3, 1, 4 ],[ 3, 2, 1 ], ...]> gap> Print(s,"\n"); [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 2 ], [ 1, 3, 4 ], [ 1, 4, 2 ], [ 1, 4, 3 ], [ 2, 1, 3 ], [ 2, 1, 4 ], [ 2, 3, 1 ], [ 2, 3, 4 ], [ 2, 4, 1 ], [ 2, 4, 3 ], [ 3, 1, 2 ], [ 3, 1, 4 ], [ 3, 2, 1 ], [ 3, 2, 4 ], [ 3, 4, 1 ], [ 3, 4, 2 ], [ 4, 1, 2 ], [ 4, 1, 3 ], [ 4, 2, 1 ], [ 4, 2, 3 ], [ 4, 3, 1 ], [ 4, 3, 2 ] ]

`‣ SurjectiveActionHomomorphismAttr` ( xset ) | ( attribute ) |

returns an action homomorphism for the external set `xset` which is surjective. (As the `Image`

(32.4-6) value of this homomorphism has to be computed to obtain the range, this may take substantially longer than `ActionHomomorphism`

(41.7-1).)

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