Preliminaries |

Let me start with some basic concepts
about *finite group actions*.
(More details can be found in [12].)
Let *G* denote a multiplicative
finite group and *X* a nonempty set.
A *finite group action* * _{G}X*
of

such thatG ´X -> X, (g,x) -> gx,

A group actiond:G -> S_{X}, g -> d(g)=:bar (g), where bar (g)(x) :=gx.

The equivalence classes are calledx ~_{G}x' iff $g ÎG : x'=gx.

The set of allG(x) := {gx | g ÎG }.

forms a partition ofG\\X:={G(x) | xÎX}

is a subgroup ofG_{x}:= {g ÎG | gx=x }

Finally the set of all fixed points of| G(x) | = | G | / | G_{x}| .

Interesting examples for group actions can be found as actions on the setX_{g}:= {x | gx=x }.

- then
*G*acts on*Y*by the definition^{X}*G´Y*^{X}-> Y^{X}, (g,f) -> f o g^{-1}, - then
*H*acts on*Y*by the definition^{X}*H´Y*^{X}-> Y^{X}, (h,f) -> h o f, - then the direct product
*G´H*acts on*Y*by the definition^{X}

De Bruijn proved that whenever a direct product*(G´H)´Y*^{X}-> Y^{X}, ((g,h),f) -> h o f o g^{-1},*G´H*is acting on a set*M*we obtain natural actions of*H*on the set of*G*-orbits:

and of*H´(G\\M) -> G\\M, (h,G(m)) -> G(hm)**G*on the set of*H*-orbits:*G´(H\\M) -> H\\M, (g,H(m)) -> H(gm).* - then
*G*acts on*X*by the definition^{X}*G´X*^{X}-> X^{X}, (g,f) -> g o f o g^{-1}, - then the
*wreath product**H wr*acts in form of the_{X}G*exponentiation*on*Y*. The wreath product is a group formed by a set^{X}*H wr*with multiplication_{X}G:={(y,g) | yÎH^{X}, gÎG}*(y,g)(y', g')=(yy'*, where_{g}, gg')*yy'*and_{g}(x):=y(x)y'_{g}(x)*y'*. It acts in a natural way on_{g}(x):=y'(g^{-1}x)*Y*by^{X}

where*(H wr*_{X}G)´Y^{X}-> Y^{X}, ((y,g),f) -> [~f] ,*[~f] (x)= y(x)f(p*. There is a bijection due to Lehmann ([13],[14]) which reduces the action of a wreath product to the action of the group^{-1}x)*G*on the set of all functions from*X*into the set of all orbits of*H*on*Y*:

where*F:H wr*_{X}G\\Y^{X}-> G\\(H\\Y)^{X}, (H wr_{X}G(f)) -> G(F),*FÎ(H\\Y)*is given by^{X}*F(x)=H(f(x))*, and*G*acts on*(H\\Y)*by^{X}*g(F):= F o g*.^{-1}

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001

Preliminaries |