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# 6.1 Ordered signatures by quotient images

Let D ⊆ G be a relative difference set with parameters (v/n,n,k,λ) and forbidden set N ⊆ G. Let U ≤ G be a normal subgroup such that U ⊆ N.

Then the coset signature (v1,...,v|G:U|) of D has only the entries 1 (k- times) and 0 (|G:U|−k- times). And as in chapter RDS:Invariants for Difference Sets we have

| ∑ j | vj vij = λ(|U|−|giU ∩N|) for gi ∉ U

where vij=|D∩gigjU|. If the forbidden set N is a subgroup of G we have |giU∩N| is either 0 or equal to |U∩N|=|U|.

Let φ:G→ G/U be the canonical epimorphism. Then Dφ is a relative difference set in G/U with forbidden set Nφ and parameters (v/n,n/|U|,k,|U|λ).

So the ordered signatures with respect to U are equivalent to the relative difference sets in G/U. Observe that we may not apply reduction in G/U using the full automorphismgroup of G/U but only the group induced by the stabiliser of U in the automorphism group of G. This is due to the fact that we use an ``induced'' notion of equivalence in G/U because we are interested in signatures and not primarily in difference sets in G/U.

- `NormalSgsForQuotientImages(` forbidden`,` Gdata`) O`

  calculates all normal subgroups of Gdata.G which lie in forbidden. The returned value is a list of normal subgroups which define pairwise non-isomorphic factor groups.

- `DataForQuotientImage(` normal`,` forbidden`,` k`,` lambda`,` Gdata`) O`

  Let Gdata be the usual record for a group G. And let k and lambda be the parameters of the relative difference set we want to find. Let then forbidden be the forbidden set (as a group or a list of group elements or integers) and normal a normal subgroup of G which is contained in forbidden.

  Then `DataForQuotientImage` returns a record containing the record .Gdata of the factor group G/U where the automorphism group is the one induced by the stabiliser of normal in the automorphism group of G. Furthermore the returned record contains the forbidden set .forbidden in G/U and the new parameter .lambda for the difference set in G/U.

  The data returned by `DataForQuotientImage` can be used to calculate difference sets in G/U in the way outlined in chapter RDS:A quick start. A quotient image of a relative difference set has a larger λ than the initial difference set. So the following invariant can be used for the generation of difference sets:

- `MultiplicityInvariantLargeLambda(` set`,` Gdata`) O`

  Let set be a partial relative difference set with λ > 1. Set ``P`:=AllPresentables(`set`,`Gdata`)` then the set of multiplicities of P is an invariant for partial relative difference sets.

  `MultiplicityInvariantLargeLambda` returns a List in a form as `Collected` does.

      gap> G:=CyclicGroup(7);;Gdata:=PermutationRepForDiffsetCalculations(G);;
      gap> AllPresentables([2,3],Gdata);
      [ 2, 3, 7, 2, 7, 6 ]
      gap> MultiplicityInvariantLargeLambda([2,3],Gdata);
      [ [ 1, 2 ], [ 2, 2 ] ]

  This invariant can be used for `ReducedStartSets` complementary to the signature invariant by defining

      gap> partfunc:=function(list)
      > local sig;
      > if sig=fail
      > then return fail;
      > fi;
      > return [MultiplicityInvariantLargeLambda(list,Gdata),SigInvariant(list,sigdata)];
      > end;
      function( list ) ... end

  and then passing partfunc to `ReducedStartSets`. Of course, sigdata has to be the list of records defining the coset signatures (see section RDS:The Coset Signature)

  After all difference sets are known, they must be converted into ordered signatures. This is done by the following function:

- `OrderedSigsFromQuotientImages(` fGroupData`,` qimages`,` forbidden`,` normal`,` Gdata`) O`

  Let Gdata be the usual record for a group G and normal a normal subgroup of G which lies in the forbidden set forbidden. Let then fGroupData be the record .Gdata describing G/normal as returned by `DataForQuotientImage` and qimages a set of difference sets in G/normal .

  Then `OrderedSigsFromQuotientImages` returns a record containing a list of ordered signatures .orderedSigs and a list of cosets .cosets as well as the factor group .fg defined by fGroupData and its full automorphism group fgaut and the image of forbidden in .fg is returned as .Nfg.

- `MatchingFGDataForOrderedSigs(` forbidden`,` Gdata`,` normalsgs`,` fgdata`) O`

  Let fgdata be a list of records of the form returned by `OrderedSigsFromQuotientImages` and normalsgs a list of normal subgroups of the group Gdata.G. Furthermore let forbidden be the forbidden set as a list of group elements or integers or a subgroup of Gdata.G.

  Then `MatchingFGDataForOrderedSigs` retruns all elements of fgdata which match a normal subgroup of normalsgs. The returned value is a record containing the normal subgroup .normal from normalsgs, the record .sigdata from fgdata and a homomorphism .hom which maps Gdata.G onto .sigdata.Gdata.G and takes forbidden to .sigdata.Nfg.

- `OrderedSigInvariant(` set`,` data`) O`

  does the same as `SigInvariant`, but for ordered signatures. Here data has to be a list of records containing ordered signatures called .orderedSigs and cosets .cosets just as returned by `OrderedSigsFromQuotientImages`.

  Assume we have calculated ordered signatures and have stored them in a record .osigs and a list normalSubgroupsData as returned by `SignatureData` containing the admissible signatures. A function for partitioning partial relative difference sets as required by `ReducedStartsets` can be defined as follows:

      partitionfunc:=function(list)
       local si, osi;
        si:=SigInvariant(Union(list,[1]),normalSubgroupsData);
        osi:=OrderedSigInvariant(Union(list,[1]),[osigs]);
        if osi=fail or si=fail
         then
          return fail;
        else
          return si;
        fi;
      end;

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