Paradigmatic ExamplesBilateral classes, symmetry classes of mappingsSylows TheoremProducts of Actions

Products of Actions

Now we take two actions into account, GX and HY, say, and derive further actions from these. Without loss of generality we can assume X ÇY= Æ since otherwise we can rename the elements of X, in order to replace GX by a similar action GX', for which X' ÇY= Æ. Now we form the (disjoint) union X È Y and let G ´H act on this set as follows:

(G ´H) ´(X È Y) -> X È Y

where ((g,h),z) -> gz if z ÎX and ((g,h),z) -> hz if z ÎY. The corresponding permutation group will be denoted by bar (G) Åbar (H) (cf. the formula) and called the direct sum of bar (G) and bar (H). Another canonical action of G ´H is that on the cartesian product:

(G ´H) ´(X ´Y) -> X ´Y :((g,h),(x,y)) -> (gx,hy).

The corresponding permutation group will be denoted by bar (G) Äbar (H) and called the cartesian product of bar (G) and bar (H). An important particular case is

Example: Assume two finite and transitive actions of G on X and Y. They yield, as was just described, a canonical action of G ´G on X ´Y which has as one of its restrictions the action of D(G ´G), the diagonal, which is isomorphic to G, on X ´Y. We notice that, for fixed x ÎX, y ÎY, the following is true (see exercise):

Hence the following is true:

Corollary: If G acts transitively on both X and Y, then, for fixed x ÎX, y ÎY, the mapping
G \\(X ´Y) -> Gx \G/Gy :G(x,gy) -> GxgGy
is a bijection (note that G \\(X ´Y) stands for D(G ´G) \\(X ´Y)). Moreover, the action of G on the orbit G(x,gy) and on the set of left cosets
G/(Gx ÇgGyg-1)
are similar. Hence, if D denotes a transversal of the set of double cosets Gx \G/Gy, for fixed x ÎX, y ÎY, then we have the following similarity:
G (X ´Y ) » G ( Èg ÎD G/(Gx ÇgGyg-1) ).

harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/
UNI-Graz Institut für Mathematik
UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

Paradigmatic ExamplesBilateral classes, symmetry classes of mappingsSylows TheoremProducts of Actions