Products 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):

• Each orbit of G on X ´Y contains an element of the form (x,gy).
• The stabilizer of (x,gy) is G_x ÇgG_yg^-1, hence the action of G on the orbit of (x,gy) is similar to the action of G on G/(G_x ÇgG_yg^-1) (recall the lemma).
• (x,gy) lies in the orbit of (x,g'y) if and only if

  G_xgG_y=G_xg'G_y.

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) -> G_x \G/G_y :G(x,gy) -> G_xgG_y

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/(G_x ÇgG_yg^-1)

are similar. Hence, if D denotes a transversal of the set of double cosets G_x \G/G_y, for fixed x ÎX, y ÎY, then we have the following similarity:

_G (X ´Y ) » _G ( È_{g ÎD} G/(G_x ÇgG_yg^-1) ).

Products of Actions