If there exists a number d(X)>0, for every point X in R^{n}, such that, for every isometry f in H which satisfies f(X)!=X, d(X,f(X))>=d(X), we say that a subgroup H of Isom(R^{n}) is discrete.

I don't understand the definition.

Also, I have to give some examples of discrete subgroups acting on R^{2}.

Can somebody help me?

Thanks