General routines |

whereN_{G}(B):={gÎG | gB=B} ,

The following two lemmata are the basic tools for recursive orbit algorithms. The first one shows how the evaluation of a transversal can be replaced by successive evaluations of blocks, their normalizers and transversals of the orbits of the normalizers on the blocks. Then the Homomorphism Principle [12] describes a method how to construct such blocks.

Lemma:Letbe a group action, and let_{G}XXbe partitioned into distinct blocksBof_{i}X,such that forX=È_{iÎI}B_{i},jÎIandgÎGthe setgB. (In other words the mapping_{j}Î{B_{i}| iÎI}B -> gBis a group action on{B.) Let_{i}| iÎI}JÍIbe such that atransversalof this action is given byThen a transversal ofT(G\\{B_{i}| iÎI} )= {B_{i}| iÎJ} .G\\Xis given bywhereT(G\\X)= È_{iÎJ}T(N_{G}(B_{i})\\B_{i}),T(Nis a transversal of_{G}(B_{i})\\B_{i})N._{G}(B_{i})\\B_{i}

Lemma:(Homomorphism Principle)Letand_{G}Xbe two finite group actions and let_{G}Yj:X -> Ybe aG-homomorphism (i.e.j(gx)=gj(x)for allgÎGandxÎX), then: The setjis a block of^{-1}({y} )={xÎX | j(x)=y}Xfor eachyÎY. The normalizer ofjis the stabilizer^{-1}({y} )Gof_{y}y.

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001

General routines |