The similitude group

Let (X,d) be a metric space and let F be the set of all bijections from X onto X such that for each f from F there exists a positive constant c_f (which may depends on f) such that
d(f(x),f(x')) = c_f * d(x,x') for any x,x' from X.

Obviously, F forms a group with respect to the functional composition operation.

At http://en.wikipedia.org/wiki/Dilation_%28mathematics%29 and at http://en.wikipedia.org/wiki/Similarity_(geometry) such functions are called as dilations and the exact similitudes, respectively (also I have seen the term "the similitude group" somewhere).

Questions:
1. What are the right terms for the group F and for its elements?
2. Can anybody give a reference on this topic?
3. Have these groups been obtained for metric spaces of majour practical use (such as Euclidean one)?
Any references will be very much appreciated.

Thanks,
Mikhail.
.

Relevant Pages

  • Re: isometries, and symmetry groups
    ... bijections which leave X invariant, ... metric space) leaving X fixed? ... Isometries are important when you are studying the ...
    (sci.math)
  • Re: isometries, and symmetry groups
    ... bijections which leave X invariant, ... metric space) leaving X fixed? ... Homeois a group regardless of the topological space X ...
    (sci.math)
Loading