46.4 Wreath Products

constructs the wreath product of the group G with the permutation group P (acting on its MovedPoints).

The second usage constructs the wreath product of the group G with the image of the group H under hom where hom must be a homomorphism from H into a permutation group. (If hom is not given, and P is not a permutation group the result of IsomorphismPermGroup(P) -- whose degree may be dependent on the method and thus is not well-defined! -- is taken for hom).

For a wreath product W of G with a permutation group P of degree n and 1 £ nr £ n the operation Embedding(W,nr) provides the embedding of G in the nr-th component of the direct product of the base group G ⁿ of W. Embedding(W,n+1) is the embedding of P into W. The operation Projection(W) provides the projection onto the acting group P (see Embeddings and Projections for Group Products).

gap> g:=Group((1,2,3),(1,2));
Group( [ (1,2,3), (1,2) ] )
gap> p:=Group((1,2,3));
Group( [ (1,2,3) ] )
gap> w:=WreathProduct(g,p);
Group( [ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8), 
  (1,4,7)(2,5,8)(3,6,9) ] )
gap> Size(w);
648
gap> Embedding(w,1);
1st embedding into Group( [ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8), 
  (1,4,7)(2,5,8)(3,6,9) ] )
gap> Image(Embedding(w,3));
Group( [ (7,8,9), (7,8) ] )
gap> Image(Embedding(w,4));
Group( [ (1,4,7)(2,5,8)(3,6,9) ] )
gap> Image(Projection(w),(1,4,8,2,6,7,3,5,9));
(1,2,3)

for two permutation groups G and H this function constructs the wreath product of G and H in the imprimitive action. If G acts on l points and H on m points this action will be on l·m points, it will be imprimitive with m blocks of size l each.

The operations Embedding and Projection operate on this product as described for general wreath products.

gap> w:=WreathProductImprimitiveAction(g,p);;
gap> LargestMovedPoint(w);
9

for two permutation groups G and H this function constructs the wreath product in product action. If G acts on l points and H on m points this action will be on l^m points.

The operations Embedding and Projection operate on this product as described for general wreath products.

gap> w:=WreathProductProductAction(g,p);
<permutation group of size 648 with 7 generators>
gap> LargestMovedPoint(w);
27

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