6 Methods for recognition

6.4 Methods for projective groups

6.4-1

6.4-2

6.4-3

6.4-4

6.4-5

6.4-6

6.4-7

6.4-8

6.4-9

6.4-10

6.4-11

6.4-12

6.4-13

6.4-14

6.4-15

6.4-16

6.4-17

6.4-18

6.4-19

6.4-20

6.4-21

6.4-22

6.4-23

6.4-24

6.4-25

6.4-26

6.4-1

`AltSymBBByDegree`

6.4-2

`BiggerScalarsOnly`

6.4-3

`BlockScalarProj`

6.4-4

`Blocks`

6.4-5

`BlocksBackToMats`

6.4-6

`BlocksModScalars`

6.4-7

`C3C5`

6.4-8

`C6`

6.4-9

`ClassicalNatural`

6.4-10

`ComputeSimpleSocle`

6.4-11

`D247`

6.4-12

`DoBaseChangeForBlocks`

6.4-13

`FindElmOfEvenNormal`

6.4-14

`KroneckerKernel`

6.4-15

`KroneckerProduct`

6.4-16

`LieTypeNonConstr`

6.4-17

`LowIndex`

6.4-18

`NameSporadic`

6.4-19

`NotAbsolutelyIrred`

6.4-20

`ProjDeterminant`

6.4-21

`PrototypeForC2C4`

6.4-22

`SporadicsByOrders`

6.4-23

`StabilizerChainProj`

6.4-24

`Subfield`

6.4-25

`TensorDecomposable`

6.4-26

`ThreeLargeElOrders`

The following methods can be equally applied to permutation, matrix and projective groups. We do not refer to them as black-box groups here, as they are allowed to contain code that only works for inputs of the listed types.

`FewGensAbelian`

This method is used for recognizing permutation groups.

If there are not too may generators (right now that means at most 200), check whether they commute; if yes, dispatch to `KnownNilpotent`

, otherwise return `NeverApplicable`

.

`KnownNilpotent`

This method is unused!

Hint to this method if you know G to be nilpotent or call it directly if you find out so. Note that it will return NeverApplicable if G is a p-group for some prime p. Make sure that the !.projective component is set correctly such that we can set the right Order method.

`SnAnUnknownDegree`

This method is unused!

This method tries to determine whether the input group given by `ri` is isomorphic to a symmetric group Sn or alternating group An with \(11 \leq n\). It is an implementation of [JLNP13].

If `Grp(ri)` is a permutation group, we assume that it is primitive and not a giant (a giant is Sn or An in natural action).

`TrivialGroup`

This method is used for recognizing permutation, matrix, and projective groups.

This method is successful if and only if all generators of a group `G` are equal to the identity. Otherwise, it returns `NeverApplicable`

indicating that it will never succeed. This method is only installed to handle the trivial case such that we do not have to take this case into account in the other methods.

The following table gives an overview over the installed methods and their rank (higher rank means higher priority, the method is tried earlier, see Chapter 4).

1250 | `FewGensAbelian` |
if very few generators, check IsAbelian and if yes, do KnownNilpotent | 6.1-1 |

1050 | `FewGensAbelian` |
if very few generators, check IsAbelian and if yes, do KnownNilpotent | 6.1-1 |

300 | `TrivialGroup` |
go through generators and compare to the identity | 6.1-4 |

100 | `ThrowAwayFixedPoints` |
try to find a huge amount of (possible internal) fixed points | 6.2-11 |

99 | `FewGensAbelian` |
if very few generators, check IsAbelian and if yes, do KnownNilpotent | 6.1-1 |

97 | `Pcgs` |
use a Pcgs to calculate a stabilizer chain | 6.2-7 |

95 | `MovesOnlySmallPoints` |
calculate a stabilizer chain if only small points are moved | 6.2-5 |

90 | `NonTransitive` |
try to find non-transitivity and restrict to orbit | 6.2-6 |

80 | `Giant` |
tries to find Sn and An in their natural actions | 6.2-2 |

70 | `Imprimitive` |
for a imprimitive permutation group, restricts to block system | 6.2-3 |

60 | `LargeBasePrimitive` |
recognises large-base primitive permutation groups | 6.2-4 |

55 | `StabilizerChainPerm` |
for a permutation group using a stabilizer chain via the genss package | 6.2-10 |

50 | `StabChain` |
for a permutation group using a stabilizer chain | 6.2-9 |

`BalTreeForBlocks`

This method is unused!

This method creates a balanced composition tree for the kernel of an imprimitive group. This is guaranteed as the method is just called from `FindHomMethodsPerm.`

`Imprimitive`

and itself. The homomorphism for the split in the composition tree used is induced by the action of `G` on half of its blocks.

`Giant`

This method is used for recognizing permutation groups.

The method tries to determine whether the input group `G` is a giant (that is, \(A_n\) or \(S_n\) in its natural action on \(n\) points). The output is either a data structure \(D\) containing nice generators for `G` and a procedure to write an SLP for arbitrary elements of `G` from the nice generators; or `NeverApplicable`

if `G` is not transitive; or `fail`

, in the case that no evidence was found that `G` is a giant, or evidence was found, but the construction of \(D\) was unsuccessful. If the method constructs \(D\) then the calling node becomes a leaf.

`Imprimitive`

This method is used for recognizing permutation groups.

If the input group is not known to be transitive then this method returns `NotEnoughInformation`

. If the input group is known to be transitive and primitive then the method returns `NeverApplicable`

; otherwise, the method tries to compute a nontrivial block system. If successful then a homomorphism to the action on the blocks is defined; otherwise, the method returns `NeverApplicable`

.

If the method is successful then it also gives a hint for the children of the node by determining whether the kernel of the action on the block system is solvable. If the answer is yes then the default value 20 for the number of random generators in the kernel construction is increased by the number of blocks.

`LargeBasePrimitive`

This method is used for recognizing permutation groups.

This method tries to determine whether the input group `G` is a fixed-point-free large-base primitive group that neither is a symmetric nor an alternating group in its natural action. This method is an implementation of [LNPS06].

A primitive group \(H\) acting on \(N\) points is called *large* if there exist \(n\), \(k\), and \(r\) with \(N = \{n \choose k\}^r\), and up to a permutational isomorphism \(H\) is a subgroup of the product action wreath product \(S_n \wr S_r\), and an overgroup of \((A_n) ^ r\) where \(S_n\) and \(A_n\) act on the \(k\)-subsets of \(\{1, ..., n\}\). This algorithm recognises fixed-point-free large primitive groups with \(r \cdot k > 1\) and \(2 \cdot r \cdot k^2 \le n\).

A large primitive group \(H\) of the above type which does have fixed points is handled as follows: if the group \(H\) does not know yet that it is primitive, then `ThrowAwayFixedPoints`

(6.2-11) returns `NotEnoughInformation`

. After the first call to `LargeBasePrimitive`

, the group \(H\) knows that it is primitive, but since it has fixed points `LargeBasePrimitive`

returns `NeverApplicable`

. Since `ThrowAwayFixedPoints`

(6.2-11) previously returned `NotEnoughInformation`

, it will be called again. Then it will use the new information about \(H\) being primitive, and is guaranteed to prune away the fixed points and set up a reduction homomorphism. `LargeBasePrimitive`

is then applicable to the image of that homomorphism.

If `G` is imprimitive then the output is `NeverApplicable`

. If `G` is primitive then the output is either a homomorphism into the natural imprimitive action of `G` on \(nr\) points with \(r\) blocks of size \(n\), or `TemporaryFailure`

, or `NeverApplicable`

if no parameters \(n\), \(k\), and \(r\) as above exist.

`MovesOnlySmallPoints`

This method is used for recognizing permutation groups.

If a permutation group moves only small points (currently, this means that its largest moved point is at most 10), then this method computes a stabilizer chain for the group via `StabChain`

. This is because the most convenient way of solving constructive membership in such a group is via a stabilizer chain. In this case, the calling node becomes a leaf node of the composition tree.

If the input group moves a large point (currently, this means a point larger than 10), then this method returns `NeverApplicable`

.

`NonTransitive`

This method is used for recognizing permutation groups.

If a permutation group `G` acts nontransitively then this method computes a homomorphism to the action of `G` on the orbit of the largest moved point. If `G` is transitive then the method returns `NeverApplicable`

.

`Pcgs`

This method is used for recognizing permutation groups.

This is the **GAP** library function to compute a stabiliser chain for a solvable permutation group. If the method is successful then the calling node becomes a leaf node in the recursive scheme. If the input group is not solvable then the method returns `NeverApplicable`

.

`PcgsForBlocks`

This method is unused!

This method is called after a hint is set in `FindHomMethodsPerm.`

`Imprimitive`

. Therefore, the group `G` preserves a non-trivial block system. This method checks whether or not the restriction of `G` on one block is solvable. If so, then `FindHomMethodsPerm.`

`Pcgs`

is called, and otherwise `NeverApplicable`

is returned.

`StabChain`

This method is used for recognizing permutation groups.

This is the randomized **GAP** library function for computing a stabiliser chain. The method selection process ensures that this function is called only with small-base inputs, where the method works efficiently.

`StabilizerChainPerm`

This method is used for recognizing permutation groups.

TODO

`ThrowAwayFixedPoints`

This method is used for recognizing permutation groups.

This method defines a homomorphism of a permutation group `G` to the action on the moved points of `G` if `G` has any fixed points, and is either known to be primitive or the ratio of fixed points to moved points exceeds a certain threshold. If `G` has fixed points but is not primitive, then it returns `NotEnoughInformation`

so that it may be called again at a later time. In all other cases, it returns `NeverApplicable`

.

In the current setup, the homomorphism is defined if the number \(n\) of moved points is at most \(1/3\) of the largest moved point of `G`, or \(n\) is at most half of the number of points on which `G` is stored internally by **GAP**.

The fact that this method returns `NotEnoughInformation`

if `G` has fixed points but does not know whether it is primitive, is important for the efficient handling of large-base primitive groups by `LargeBasePrimitive`

(6.2-4).

The following table gives an overview over the installed methods and their rank (higher rank means higher priority, the method is tried earlier, see Chapter 4). Note that there are not that many methods for matrix groups since the system can switch to projective groups by dividing out the subgroup of scalar matrices. The bulk of the recognition methods are then installed es methods for projective groups.

3100 | `TrivialGroup` |
go through generators and compare to the identity | 6.1-4 |

1175 | `KnownStabilizerChain` |
use an already known stabilizer chain for this group | 6.3-6 |

1100 | `DiagonalMatrices` |
check whether all generators are diagonal matrices | 6.3-4 |

1000 | `ReducibleIso` |
use the MeatAxe to find invariant subspaces | 6.3-9 |

900 | `GoProjective` |
divide out scalars and recognise projectively | 6.3-5 |

`BlockDiagonal`

This method is unused!

This method is only called when a hint was passed down from the method `BlockLowerTriangular`

. In that case, it knows that the group is in block diagonal form. The method is used both in the matrix- and the projective case.

The method immediately delegates to projective methods handling all the diagonal blocks projectively. This is done by giving a hint to the image to use the method `BlocksModScalars`

. The method for the kernel then has to deal with only scalar blocks, either projectively or with scalars, which is again done by giving a hint to either use `BlockScalar`

or `BlockScalarProj`

respectively.

Note that this method is implemented in a way such that it can also be used as a method for a projective group `G`. In that case the recognition node has the `!.projective`

component bound to `true`

and this information is passed down to image and kernel.

`BlockLowerTriangular`

This method is unused!

This method is only called when a hint was passed down from the method `ReducibleIso`

. In that case, it knows that a base change to block lower triangular form has been performed. The method can then immediately find a homomorphism by mapping to the diagonal blocks. It sets up this homomorphism and gives hints to image and kernel. For the image, the method `BlockDiagonal`

is used and for the kernel, the method `LowerLeftPGroup`

is used.

Note that this method is implemented in a way such that it can also be used as a method for a projective group `G`. In that case the recognition node has the `!.projective`

component bound to `true`

and this information is passed down to image and kernel.

`BlockScalar`

This method is unused!

This method is only called by a hint. Alongside with the hint it gets a block decomposition respected by the matrix group `G` to be recognised and the promise that all diagonal blocks of all group elements will only be scalar matrices. This method recursively builds a balanced tree and does scalar recognition in each leaf.

`DiagonalMatrices`

This method is used for recognizing matrix groups.

This method is successful if and only if all generators of a matrix group `G` are diagonal matrices. Otherwise, it returns `NeverApplicable`

.

`GoProjective`

This method is used for recognizing matrix groups.

This method defines a homomorphism from a matrix group `G` into the projective group `G` modulo scalar matrices. In fact, since projective groups in **GAP** are represented as matrix groups, the homomorphism is the identity mapping and the only difference is that in the image the projective group methods can be applied. The bulk of the work in matrix recognition is done in the projective group setting.

`KnownStabilizerChain`

This method is used for recognizing matrix groups.

If a stabilizer chain is already known, then the kernel node is given knowledge about this known stabilizer chain, and the image node is told to use homomorphism methods from the database for permutation groups. If a stabilizer chain of a parent node is already known this is used for the computation of a stabilizer chain of this node. This stabilizer chain is then used in the same way as above.

`LowerLeftPGroup`

This method is unused!

This method is only called by a hint from `BlockLowerTriangular`

as the kernel of the homomorphism mapping to the diagonal blocks. The method uses the fact that this kernel is a \(p\)-group where \(p\) is the characteristic of the underlying field. It exploits this fact and uses this special structure to find nice generators and a method to express group elements in terms of these.

`NaturalSL`

This method is unused!

TODO

`ReducibleIso`

This method is used for recognizing matrix and projective groups.

This method determines whether a matrix group `G` acts irreducibly. If yes, then it returns `NeverApplicable`

. If `G` acts reducibly then a composition series of the underlying module is computed and a base change is performed to write `G` in a block lower triangular form. Also, the method passes a hint to the image group that it is in block lower triangular form, so the image immediately can make recursive calls for the actions on the diagonal blocks, and then to the lower \(p\)-part. For the image the method `BlockLowerTriangular`

is used.

Note that this method is implemented in a way such that it can also be used as a method for a projective group `G`. In that case the recognition node has the `!.projective`

component bound to `true`

and this information is passed down to image and kernel.

`Scalar`

This method is unused!

TODO

The following table gives an overview over the installed methods and their rank (higher rank means higher priority, the method is tried earlier, see Chapter 4). Note that the recognition for matrix group switches to projective recognition rather soon in the recognition process such that most recognition methods in fact are installed as methods for projective groups.

3000 | `TrivialGroup` |
go through generators and compare to the identity | 6.1-4 |

1300 | `ProjDeterminant` |
find homomorphism to non-zero scalars mod d-th powers | 6.4-20 |

1200 | `ReducibleIso` |
use the MeatAxe to find invariant subspaces | 6.3-9 |

1100 | `NotAbsolutelyIrred` |
write over a bigger field with smaller degree | 6.4-19 |

1050 | `ClassicalNatural` |
check whether it is a classical group in its natural representation | 6.4-9 |

1000 | `Subfield` |
write over a smaller field with same degree | 6.4-24 |

900 | `C3C5` |
compute a normal subgroup of derived and resolve C3 and C5 | 6.4-7 |

850 | `C6` |
find either an (imprimitive) action or a symplectic one | 6.4-8 |

840 | `D247` |
play games to find a normal subgroup | 6.4-11 |

810 | `AltSymBBByDegree` |
try BB recognition for dim+1 and/or dim+2 if sensible | 6.4-1 |

800 | `TensorDecomposable` |
find a tensor decomposition | 6.4-25 |

700 | `FindElmOfEvenNormal` |
find D2, D4 or D7 by finding an element of an even normal subgroup | 6.4-13 |

600 | `LowIndex` |
find an (imprimitive) action on subspaces | 6.4-17 |

580 | `NameSporadic` |
generate maximal orders | 6.4-18 |

550 | `ComputeSimpleSocle` |
compute simple socle of almost simple group | 6.4-10 |

500 | `ThreeLargeElOrders` |
recognise Lie type groups and get its characteristic | 6.4-26 |

400 | `LieTypeNonConstr` |
do non-constructive recognition of Lie type groups | 6.4-16 |

100 | `StabilizerChainProj` |
last resort: compute a stabilizer chain (projectively) | 6.4-23 |

`AltSymBBByDegree`

This method is used for recognizing projective groups.

This method is a black box constructive (?) recognition of alternating and symmetric groups.

This algorithm is probably based on the paper [BLGN+05].

`BiggerScalarsOnly`

This method is unused!

TODO

`BlockScalarProj`

This method is unused!

This method is only called by a hint. Alongside with the hint it gets a block decomposition respected by the matrix group `G` to be recognised and the promise that all diagonal blocks of all group elements will only be scalar matrices. This method simply norms the last diagonal block in all generators by multiplying with a scalar and then delegates to `BlockScalar`

(see 6.3-3) and matrix group mode to do the recognition.

`Blocks`

This method is unused!

TODO

`BlocksBackToMats`

This method is unused!

TODO

`BlocksModScalars`

This method is unused!

This method is only called when hinted from above. In this method it is understood that G should *neither* be recognised as a matrix group *nor* as a projective group. Rather, it treats all diagonal blocks modulo scalars which means that two matrices are considered to be equal, if they differ only by a scalar factor in *corresponding* diagonal blocks, and this scalar can be different for each diagonal block. This means that the kernel of the homomorphism mapping to a node which is recognised using this method will have only scalar matrices in all diagonal blocks.

This method does the balanced tree approach mapping to subsets of the diagonal blocks and finally using projective recognition to recognise single diagonal block groups.

`C3C5`

This method is used for recognizing projective groups.

TODO

`C6`

This method is used for recognizing projective groups.

This method is designed for the handling of the Aschbacher class C6 (normaliser of an extraspecial group). If the input `G`\(\le PGL(d,q)\) does not satisfy \(d=r^n\) and \(r|q-1\) for some prime \(r\) and integer \(n\) then the method returns `NeverApplicable`

. Otherwise, it returns either a homomorphism of `G` into \(Sp(2n,r)\), or a homomorphism into the C2 permutation action of `G` on a decomposition of \(GF(q)^d\), or `fail`

.

`ClassicalNatural`

This method is used for recognizing projective groups.

TODO

`ComputeSimpleSocle`

This method is used for recognizing projective groups.

This method randomly computes the non-abelian simple socle and stores it along with additional information if it is called for an almost simple group. Once the non-abelian simple socle is computed the function does not need to be called again for this node and therefore returns `NeverApplicable`

.

`D247`

This method is used for recognizing projective groups.

TODO

`DoBaseChangeForBlocks`

This method is unused!

TODO

`FindElmOfEvenNormal`

This method is used for recognizing projective groups.

TODO

`KroneckerKernel`

This method is unused!

TODO

`KroneckerProduct`

This method is unused!

TODO

`LieTypeNonConstr`

This method is used for recognizing projective groups.

Recognise quasi-simple group of Lie type when characteristic is given. Based on [BKPS02] and [AB01].

`LowIndex`

This method is used for recognizing projective groups.

This method is designed for the handling of the Aschbacher class C2 (stabiliser of a decomposition of the underlying vector space), but may succeed on other types of input as well. Given `G` \( \le PGL(d,q)\), the output is either the permutation action of `G` on a short orbit of subspaces or `fail`

. In the current setup, "short orbit" is defined to have length at most \(4d\).

`NameSporadic`

This method is used for recognizing projective groups.

This method returns a list of sporadic simple groups that the group underlying `ri` could be. It does not recognise extensions of sporadic simple groups nor the Monster and the Baby Monster group. It is based on the Magma v2.24.10 function `RecognizeSporadic`

.

`NotAbsolutelyIrred`

This method is used for recognizing projective groups.

If an irreducible projective group `G` acts absolutely irreducibly then this method returns `NeverApplicable`

. If `G` is not absolutely irreducible then a homomorphism into a smaller dimensional representation over an extension field is defined. A hint is handed down to the image that no test for absolute irreducibility has to be done any more. Another hint is handed down to the kernel indicating that the only possible kernel elements can be elements in the centraliser of `G` in \(PGL(d,q)\) that come from scalar matrices in the extension field.

`ProjDeterminant`

This method is used for recognizing projective groups.

The method defines a homomorphism from a projective group `G`\( \le PGL(d,q)\) to the cyclic group \(GF(q)^*/D\), where \(D\) is the set of \(d\)th powers in \(GF(q)^*\). The image of a group element \(g \in \textit{G}\) is the determinant of a matrix representative of \(g\), modulo \(D\).

`PrototypeForC2C4`

This method is unused!

TODO/FIXME: PrototypeForC2C4 is not used anywhere

`SporadicsByOrders`

This method is unused!

This method returns a list of sporadic simple groups that `G` possibly could be. Therefore it checks whether `G` has elements of orders that do not appear in sporadic groups and otherwise checks whether the most common ("killer") orders of the sporadic groups appear. Afterwards it creates hints that come out of a table for the sporadic simple groups.

`StabilizerChainProj`

This method is used for recognizing projective groups.

This method computes a stabiliser chain and a base and strong generating set using projective actions. This is a last resort method since for bigger examples no short orbits can be found in the natural action. The strong generators are the nice generator in this case and expressing group elements in terms of the nice generators ist just sifting along the stabiliser chain.

`Subfield`

This method is used for recognizing projective groups.

TODO

`TensorDecomposable`

This method is used for recognizing projective groups.

TODO/FIXME: it is unclear if the following description actually belongs to this method, so be cautious!

This method currently tries to find one tensor factor by powering up commutators of random elements to elements of prime order. This seems to work quite well provided that the two tensor factors are not "linked" too much such that there exist enough elements that act with different orders on both tensor factors.

This method and its description needs some improvement.

`ThreeLargeElOrders`

This method is used for recognizing projective groups.

In the case when the input group `G`\( \le PGL(d,p^e)\) is suspected to be simple but not alternating, this method takes the three largest element orders from a sample of pseudorandom elements of `G`. From these element orders, it tries to determine whether `G` is of Lie type and the characteristic of `G` if it is of Lie type. In the case when `G` is of Lie type of characteristic different from \(p\), the method also provides a short list of the possible isomorphism types of `G`.

This method assumes that its input is neither alternating nor sporadic and that `ComputeSimpleSocle`

(6.4-10) has already been called.

This recognition method is based on the paper [KS09].

The following table gives an overview over the methods which are currently unused.

`KnownNilpotent` |
`FindHomMethodsGeneric` |
6.1-2 |

`SnAnUnknownDegree` |
`FindHomMethodsGeneric` |
6.1-3 |

`PcgsForBlocks` |
`FindHomMethodsPerm` |
6.2-8 |

`BalTreeForBlocks` |
`FindHomMethodsPerm` |
6.2-1 |

`BlockScalar` |
`FindHomMethodsMatrix` |
6.3-3 |

`NaturalSL` |
`FindHomMethodsMatrix` |
6.3-8 |

`Scalar` |
`FindHomMethodsMatrix` |
6.3-10 |

`BlockLowerTriangular` |
`FindHomMethodsMatrix` |
6.3-2 |

`BlockDiagonal` |
`FindHomMethodsMatrix` |
6.3-1 |

`LowerLeftPGroup` |
`FindHomMethodsMatrix` |
6.3-7 |

`DoBaseChangeForBlocks` |
`FindHomMethodsProjective` |
6.4-12 |

`Blocks` |
`FindHomMethodsProjective` |
6.4-4 |

`BlocksModScalars` |
`FindHomMethodsProjective` |
6.4-6 |

`BlocksBackToMats` |
`FindHomMethodsProjective` |
6.4-5 |

`KroneckerProduct` |
`FindHomMethodsProjective` |
6.4-15 |

`KroneckerKernel` |
`FindHomMethodsProjective` |
6.4-14 |

`BiggerScalarsOnly` |
`FindHomMethodsProjective` |
6.4-2 |

`PrototypeForC2C4` |
`FindHomMethodsProjective` |
6.4-21 |

`SporadicsByOrders` |
`FindHomMethodsProjective` |
6.4-22 |

`BlockScalarProj` |
`FindHomMethodsProjective` |
6.4-3 |

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