47.8 Primitive Permutation Groups

GAP contains a library of primitive permutation groups which includes the following permutation groups up to permutation isomorphism (i.e., up to conjugacy in the corresponding symmetric group)

·
the non-affine primitive permutation groups of degree < = 999, described in DixonMortimer88, with generators calculated in Theissen97,
·
all primitive permutation groups of degree < 256, in particular,
   °
the primitive permutation groups up to degree 50, calculated by C. Sims,
   °
the solvable (hence affine) primitive permutation groups of degree < 256, calculated by M. Short Sho92,
   °
the insolvable affine primitive permutation groups of degree < 256, calculated in Theissen97.
Note that the affine primitive permutation groups of degrees 256--999 are not included.

For degree up to 50, the names used are as given by Buekenhout and Leemans BuekenhoutLeemans96.

The names for the groups of higher degree are as chosen by Theissen97 and reflect the cohort structure in DixonMortimer88. They do not conform to the usual naming for extensions. That is l3,4.3 is the third (in some arbitrary ordering) group in a cohort for socle l3,4, but the socle factor is not necessarily of order 3.

As the work in Theissen97 has not been checked independently for completeness, it is perceivable in theory that for degrees above 50 groups are missing. However when preparing the library it has been ensured that the groups in it are all primitive and not conjugate.

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 Sn-representative, the generating set or the name for eternity.
·
Different groups in the library are not conjugate in Sn.
·
If a group in the library has a primitive subgroup with the same socle, this group is in the library as well.

  • PrimitiveGroup( deg, nr ) F

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

    The arrangement of the groups differs from the arrangement of primitive groups in the list of C. Sims, which was used in GAP 3. See SimsNo (SimsNo).

  • NrPrimitiveGroups( deg ) F

    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_5xA_5):2

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

    NrMovedPoints, Size, Transitivity, ONanScottType, IsSimpleGroup, IsSolvableGroup, and SocleTypePrimitiveGroup.

    (Note, that for groups of degree up to 999, ONan-Scott types 4a, 4b and 5 cannot occur.)

  • PrimitiveGroupsIterator( attr1, val1, attr2, val2, ... ) F

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

  • PrimitiveIdentification( G ) A

    For a primitive permutation group for which an Sn-conjugate exists in the library of primitive permutation groups (see Primitive Permutation Groups), 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. 

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

  • SimsNo( G ) A

    If G is a primitive group obtained by PrimitiveGroup (respectively one of the selection functions) 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

  • COHORTS_PRIMITIVE_GROUPS V

    In DixonMortimer88 the primitive groups are sorted in ``cohorts'' according to their socle. The variable COHORTS_PRIMITIVE_GROUPS contains for each degree 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, section 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 \F72, L2(7)2, A72 and A49 respectively. the group PrimitiveGroup(49,36), which is isomorphic to (A7×A7):22, lies in the third cohort with socle (A7×A7). 

    gap> COHORTS_PRIMITIVE_GROUPS[49];
[ [ rec( series := "Z", width := 2, parameter := 7 ), 
      [ 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( series := "L", width := 2, parameter := [ 2, 7 ] ), [ 34 ] ], 
  [ rec( series := "A", width := 2, parameter := 7 ), [ 35, 36, 37, 38 ] ], 
  [ rec( series := "A", parameter := 49, width := 1 ), [ 39, 40 ] ] ]

  • IrreducibleSolvableGroup( n, p, i ) F

    returns the i-th irreducible solvable subgroup of GL( n, p ). The irreducible solvable subgroups of GL(n,p) are ordered with respect to the following criteria:

    -
    increasing size;
    -
    increasing guardian number.
    If two groups have the same size and guardian, they are in no particular order. (See the library documentation or Sho92 for the meaning of guardian.)

    The corresponding selection functions are AllIrreducibleSolvableGroups and OneIrreducibleSolvableGroup. They obtain the following properties from the database without having to compute them anew: Dimension for the linear degree, CharFFE for the field characteristic, Size, IsLinearlyPrimitive and MinimalBlockDimension.

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    GAP 4 manual
    February 2000