Bilateral classes, symmetry classes of mappings



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Bilateral classes, symmetry classes of mappings

From a given action we can derive various other actions in a natural way, e.g. yields , being the homomorphic image of in , which was already mentioned. We also obtain the subactions   on subsets which are nonempty unions of orbits. Furthermore there are the restrictions to the subgroups of . As the orbits of are unions of orbits of , the comparison of actions and restrictions is a suitable way of generalizing or specializing structures if they can be defined as orbits. The following example will show what is meant by this.

. Example   Let denote a subgroup of the direct product . Then U acts on as follows:

The orbits of this action are called the bilateral classes   of with respect to . By specializing we obtain various interesting group theoretical structures some of which have been mentioned already:

Hence left and right cosets, double cosets and conjugacy classes turn out to be special cases of bilateral classes. Being orbits, two of them are either equal or disjoint, moreover, their order is the index of the stabilizer of an element. We have mentioned this in connection with conjugacy classes and centralizers of elements, here is the consequence for double cosets: Since

we obtain



next up previous contents
Next: Double Cosets Up: Actions Previous: Similar Actions



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