GAP (primgrp) - Chapter 1: Primitive Permutation Groups

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1 Primitive Permutation Groups
1.1 Primitive Permutation Groups
1.1-1 PrimitiveGroup
1.1-2 NrPrimitiveGroups
1.1-3 AllPrimitiveGroups
1.1-4 OnePrimitiveGroup
1.1-5 PrimitiveGroupsIterator
1.1-6 COHORTS_PRIMITIVE_GROUPS
1.2 Index numbers of primitive groups
1.2-1 PrimitiveIdentification
1.2-2 SimsNo
1.2-3 PRIMITIVE_INDICES_MAGMA

### 1 Primitive Permutation Groups

#### 1.1 Primitive Permutation Groups

GAP contains a library of primitive permutation groups which includes, up to permutation isomorphism (i.e., up to conjugacy in the corresponding symmetric group), all primitive permutation groups of degree \(< 4096\), calculated in [RD05] and [Qui11], in particular,

- the primitive permutation groups up to degree 50, calculated by C. Sims,

- the primitive groups with insoluble socles of degree \(< 1000\) as calculated in [DM88],

- the solvable (hence affine) primitive permutation groups of degree \(< 256\) as calculated by M. Short [Sho92],

- some insolvable affine primitive permutation groups of degree \(< 256\) as calculated in [The97].

- The solvable primitive groups of degree up to \(999\) as calculated in [EH03].

- The primitive groups of affine type of degree up to \(999\) as calculated in [RDU03].

Not all groups are named, those which do have names use ATLAS notation. Not all names are necessary unique!

The list given in [RD05] is believed to be complete, correcting various omissions in [DM88], [Sho92] and [The97].

In detail, we guarantee the following properties for this and further versions (but not versions which came before GAP 4.2) of the library:

- All groups in the library are primitive permutation groups of the indicated degree.

- The positions of the groups in the library are stable. That is `PrimitiveGroup(n,nr)` will always give you a permutation isomorphic group. Note however that we do not guarantee to keep the chosen \(S_n\)-representative, the generating set or the name for eternity.

- Different groups in the library are not conjugate in \(S_n\).

- If a group in the library has a primitive subgroup with the same socle, this group is in the library as well.

(Note that the arrangement of groups is not guaranteed to be in increasing size, though it holds for many degrees.)

The selection functions (see Reference: Selection Functions) for the primitive groups library are `AllPrimitiveGroups` and `OnePrimitiveGroup`. They obtain the following properties from the database without having to compute them anew:

`NrMovedPoints` (Reference: NrMovedPoints for a list or collection of permutations), `Size` (Reference: Size), `Transitivity` (Reference: Transitivity for a group and an action domain), `ONanScottType` (Reference: ONanScottType), `IsSimpleGroup` (Reference: IsSimpleGroup), `IsSolvableGroup` (Reference: IsSolvableGroup), and `SocleTypePrimitiveGroup` (Reference: SocleTypePrimitiveGroup).

(Note, that for groups of degree up to 2499, O'Nan-Scott types 4a, 4b and 5 cannot occur.)

##### 1.1-1 PrimitiveGroup

`‣ PrimitiveGroup`( deg, nr ) | ( function )

returns the primitive permutation group of degree deg with number nr from the list.

The arrangement of the groups of degrees not greater than 50 differs from the arrangement of primitive groups in the list of C. Sims, which was used in GAP 3. See `SimsNo` (1.2-2).

##### 1.1-2 NrPrimitiveGroups

`‣ NrPrimitiveGroups`( deg ) | ( function )

returns the number of primitive permutation groups of degree deg in the library.

    gap> NrPrimitiveGroups(25);
    28
    gap> PrimitiveGroup(25,19);
    5^2:((Q(8):3)'4)
    gap> PrimitiveGroup(25,20);
    ASL(2, 5)
    gap> PrimitiveGroup(25,22);
    AGL(2, 5)
    gap> PrimitiveGroup(25,23);
    (A(5) x A(5)):2

##### 1.1-3 AllPrimitiveGroups

`‣ AllPrimitiveGroups`( attr1, val1, attr2, val2, ... ) | ( function )

This is a selection function which permits to select all groups from the Primitive Group Library that have a given set of properties. It accepts arguments as specified in Section Reference: Selection Functions of the GAP reference manual.

##### 1.1-4 OnePrimitiveGroup

`‣ OnePrimitiveGroup`( attr1, val1, attr2, val2, ... ) | ( function )

This is a selection function which permits to select at most one group from the Primitive Group Library that have a given set of properties. It accepts arguments as specified in Section Reference: Selection Functions of the GAP reference manual.

##### 1.1-5 PrimitiveGroupsIterator

`‣ PrimitiveGroupsIterator`( attr1, val1, attr2, val2, ... ) | ( function )

returns an iterator through `AllPrimitiveGroups(attr1,val1,attr2,val2,...)` without creating all these groups at the same time.

##### 1.1-6 COHORTS_PRIMITIVE_GROUPS

`‣ COHORTS_PRIMITIVE_GROUPS` | ( global variable )

In [DM88] the primitive groups are sorted in "cohorts" according to their socle. For each degree less than 2500, the variable `COHORTS_PRIMITIVE_GROUPS` contains a list of the cohorts for the primitive groups of this degree. Each cohort is represented by a list of length 2, the first entry specifies the socle type (see `SocleTypePrimitiveGroup` (Reference: SocleTypePrimitiveGroup)), the second entry listing the index numbers of the groups in this degree.

For example in degree 49, we have four cohorts with socles \((ℤ / 7 ℤ)^2\), \(L_2(7)^2\), \(A_7^2\) and \(A_{49}\) respectively. the group `PrimitiveGroup(49,36)`, which is isomorphic to \((A_7 \times A_7):2^2\), lies in the third cohort with socle \((A_7 \times A_7)\).

    gap> COHORTS_PRIMITIVE_GROUPS[49];
    [ [ rec( parameter := 7, series := "Z", width := 2 ),
          [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
              20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33 ] ],
      [ rec( parameter := [ 2, 7 ], series := "L", width := 2 ), [ 34 ] ],
      [ rec( parameter := 7, series := "A", width := 2 ), [ 35, 36, 37, 38 ] ],
      [ rec( parameter := 49, series := "A", width := 1 ), [ 39, 40 ] ] ]

#### 1.2 Index numbers of primitive groups

##### 1.2-1 PrimitiveIdentification

`‣ PrimitiveIdentification`( G ) | ( attribute )

For a primitive permutation group for which an \(S_n\)-conjugate exists in the library of primitive permutation groups (see 1.1), this attribute returns the index position. That is G is conjugate to `PrimitiveGroup(NrMovedPoints(G),PrimitiveIdentification(G))`.

Methods only exist if the primitive groups library is installed.

Note: As this function uses the primitive groups library, the result is only guaranteed to the same extent as this library. If it is incomplete, `PrimitiveIdentification` might return an existing index number for a group not in the library.

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

##### 1.2-2 SimsNo

`‣ SimsNo`( G ) | ( attribute )

If G is a primitive group of degree not greater than 50, obtained by `PrimitiveGroup` (1.1-1) (respectively one of the selection functions), then this attribute contains the number of the isomorphic group in the original list of C. Sims. (This is the arrangement as it was used in GAP 3.)

    gap> g:=PrimitiveGroup(25,2);
    5^2:S(3)
    gap> SimsNo(g);
    3

As mentioned in the previous section, the index numbers of primitive groups in GAP are guaranteed to remain stable. (Thus, missing groups will be added to the library at the end of each degree.) In particular, it is safe to refer to a primitive group of type deg, nr in the GAP library.

##### 1.2-3 PRIMITIVE_INDICES_MAGMA

`‣ PRIMITIVE_INDICES_MAGMA` | ( global variable )

The system Magma also provides a list of primitive groups (see [RDU03]). For historical reasons, its indexing up to degree 999 differs from the one used by GAP. The variable `PRIMITIVE_INDICES_MAGMA` can be used to obtain this correspondence. The magma index number of the GAP group `PrimitiveGroup(deg,nr)` is stored in the entry `PRIMITIVE_INDICES_MAGMA[deg][nr]`, for degree at most 999.

Vice versa, the group of degree deg with Magma index number nr has the GAP index

`Position(PRIMITIVE_INDICES_MAGMA[deg],nr)`, in particular it can be obtained by the GAP command

`PrimitiveGroup(deg,Position(PRIMITIVE_INDICES_MAGMA[deg],nr));`

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