40.4 Primitive Groups

returns the type of G of a primitiev permutation group G, according to the O'Nan-Scott classification. The labelling of the different types is not consistent in the literature, we use the following:

1

Affine.

2

Almost simple.

3a

Diagonal, Socle consists of two normal subgroups.

3b

Diagonal, Socle is minimal normal.

4a

Product action with the first factor primitive of type 3a.

4b

Product action with the first factor primitive of type 3b.

4c

Product action with the first factor primitive of type 2.

5

Twisted wreath product

If G is not a permutation group or does not act primitively on the points moved by it, the result is undefined.

returns the socle type of a primitive permutation group. The socle of a primitive group is the direct product of isomorphic simple groups, therefore the type is indicated by a record with components series, parameter (both as described under IsomorphismTypeFiniteSimpleGroup, see IsomorphismTypeFiniteSimpleGroup) and width for the number of direct factors.

If G does not have a faithful primitive action, the result is undefined.

gap> g:=AlternatingGroup(5);;
gap> h:=DirectProduct(g,g);;
gap> p:=List([1,2],i->Projection(h,i));;
gap> ac:=Action(h,AsList(g),                                                   
> function(g,h) return Image(p[1],h)^-1*g*Image(p[2],h);end);;                 
gap> Size(ac);NrMovedPoints(ac);IsPrimitive(ac,[1..60]);
3600
60
true
gap> ONanScottType(ac);
"3a"
gap> SocleTypePrimitiveGroup(ac);
rec( series := "A", width := 2, 
  name := "A(5) ~ A(1,4) = L(2,4) ~ B(1,4) = O(3,4) ~ C(1,4) = S(2,4) ~ 2A(1,4\
) = U(2,4) ~ A(1,5) = L(2,5) ~ B(1,5) = O(3,5) ~ C(1,5) = S(2,5) ~ 2A(1,5) = U\
(2,5)", parameter := 5 )

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