Paradigmatic Examples |

Finally we introduce the actions derived from * _{G}X* and

Y^{X}:= {f | f:X -> Y },

and notice:
If *G* acts on *X* and *H* acts on *Y*, then
*G*, *H* and *H ´G* act on *Y ^{X}* as follows:

*G ´Y*, i.e.^{X}-> Y^{X}:(g,f) -> f o bar (g)^{-1}*(g,f)*is mapped onto*[~f]*, where*[~f] (x):=f(g*. The corresponding permutation group on^{-1}x)*Y*will be denoted by^{X}*E*.^{ bar (G)}*H ´Y*i.e.^{X}-> Y^{X}:(h,f) -> bar (h) o f,*(h,f)*is mapped onto*[~f]*, where*[~f] (x):=hf(x)*. The corresponding permutation group will be denoted by*bar (H)*.^{E}*(H ´G) ´Y*, i.e.^{X}-> Y^{X}:((h,g),f) -> bar (h) o f o bar (g)^{-1}*((h,g),f)*is mapped onto*[~f]*, where*[~f] (x):=hf(g*. The corresponding permutation group on^{-1}x)*Y*will be denoted by^{X}*bar (H)*, and it will be called the^{ bar (G)}*power group*of*bar (H)*by*bar (G)*.

There is a fourth action which contains these three actions as subactions,
but in order to describe it we first need to introduce the *wreath product*
* H wr _{X} G *:
Its underlying set is

H wr_{X}G :=H^{X}´G= {( y,g) | y:X -> H,g ÎG },

and the multiplication is defined by

( y,g)( y',g'):= ( yy'_{g},gg'), yy'_{g}(x):= y(x) y'_{g}(x):= y(x) y'(g^{-1}x).

The actions * _{G}X* and

H wr_{X}G ´Y^{X}-> Y^{X}:(( y,g),f) -> [~f] , [~f] (x):= y(x)f(g^{-1}x).

The corresponding permutation group
on *Y ^{X}* will be denoted by

A few remarks concerning the wreath
product * H wr _{X} G * are in order, they
will in particular show that the actions of

Lemma:The wreath productH wrhas the following properties:_{X}G

- The identity element of
H wris_{X}G( i,1), wherei:x -> 1._{H}- If we define
yby^{-1}ÎH^{X}y, we get^{-1}(x):= y(x)^{-1}( y,g)^{-1}=( y^{-1}_{g-1},g^{-1}), where y^{-1}_{g-1}:= ( y^{-1})_{g-1}=( y_{g-1})^{-1}.- The normal subgroup
is called theH^{*}:= {( y,1) | yÎH^{X}} lefttriangleeq H wr_{X}G ,base group, and it is the direct product of| X |copiesHof^{x}H:H^{x}:= {( y,1) | " x' not = x : y(x')=1_{H}} simeq H, for each x ÎX.- The subgroup
G ':= {( i,g) | g ÎG } simeq Gis a complement ofH, so that we have^{*}H wr_{X}G =H^{*}·G ' , H^{*}lefttriangleeq H wr_{X}G , H^{*}ÇG ' = {( i,1) }.- The diagonal
satisfiesD(H^{*}):= {( y,1) | y constant } simeq H,D(H^{*}) ·G ' = {( y,g) | y constant, g ÎG } simeq H ´G.

This shows that the subgroups *G '*, * D(H ^{*})* and

G hookrightarrow H wr_{X}G , H hookrightarrow H wr_{X}G , H ´G hookrightarrow H wr_{X}G ,

so that in fact the actions of *G*, *H*, and *H ´G* on *Y ^{X}*
introduced above are restrictions of the
action of

In order to prepare later applications of such actions we mention that actions of direct products and of wreath products can be reformulated in terms of the respective factors of the direct and of the wreath product. Here is, to begin with, a lemma on actions of direct products, which is very easy to check (exercise):

harald.fripertinger "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

Paradigmatic Examples |