   General routines

# General routines

A subset BÍX is called a block of X if and only if for each gÎG either gB=B or gBÇB=Æ holds. The normalizer of B is defined as
NG(B):={gÎG | gB=B} ,
where gB:={gb | bÎB} .

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  describes a method how to construct such blocks.

Lemma: Let GX be a group action, and let X be partitioned into distinct blocks Bi of X,
X=ÈiÎIBi,
such that for jÎI and gÎG the set gBjÎ{Bi | iÎI} . (In other words the mapping B -> gB is a group action on {Bi | iÎI} .) Let JÍI be such that a transversal of this action is given by
T(G\\{Bi | iÎI} )= {Bi | iÎJ} .
Then a transversal of G\\X is given by
T(G\\X)= ÈiÎJT(NG(Bi)\\Bi),
where T(NG(Bi)\\Bi) is a transversal of NG(Bi)\\Bi.
Lemma: (Homomorphism Principle) Let GX and GY be two finite group actions and let j:X -> Y be a G-homomorphism (i.e. j(gx)=gj(x) for all gÎG and xÎX), then: The set j-1({y} )={xÎX | j(x)=y} is a block of X for each yÎY. The normalizer of j-1({y} ) is the stabilizer Gy of y.

harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001   General routines