Products of Actions



next up previous contents
Next: Paradigmatic Examples Up: Actions Previous: Sylows Theorem

Products of Actions

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

 

The corresponding permutation group will be denoted by (cf. gif) and called the direct sum  of and . Another canonical action of is that on the cartesian product:

 

The corresponding permutation group will be denoted by and called the cartesian product   of and . An important particular case is

. Example   Assume two finite and transitive actions of on and . They yield, as was just described, a canonical action of on which has as one of its restrictions the action of , the diagonal, which is isomorphic to , on . We notice that, for fixed , the following is true (exercise gif):

Hence the following is true:

. Corollary   If acts transitively on both and , then, for fixed , the mapping

is a bijection (note that stands for ). Moreover, the action of on the orbit and on the set of left cosets

are similar. Hence, if denotes a transversal of the set of double cosets for fixed , then we have the following similarity:

Exercises

E .   Prove the statements of gif.



next up previous contents
Next: Paradigmatic Examples Up: Actions Previous: Sylows Theorem



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995