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 [12] 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
