Paradigmatic Examples |

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

*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)*.

and the multiplication is defined byH wr_{X}G :=H^{X}´G= {( y,g) | y:X -> H,g ÎG },

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

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

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

so that in fact the actions ofG hookrightarrow H wr_{X}G , H hookrightarrow H wr_{X}G , H ´G hookrightarrow H wr_{X}G ,

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@kfunigraz.ac.at,

last changed: August 28, 2001

Paradigmatic Examples |