Topological transformation groups on hyperspace and Dugundji compacta

For a given topological transformations group on a Tychonoff space a topological transformations group is constructed on the corresponding hyperspace. We show that if the action of the given group is open, then the induced action is also open. In this case an example is built showing that the openness of the given action is significant. Further we establish that if the diagonal product of a given family of continuous mappings of Tychonoff space is an embedding, then the diagonal product of the family of induced maps of the corresponding hyperspace is also an embedding. One criteria of the Dugundji compactness of hyperspace is obtained.