Semigroup of transformations preserving a length

Let X_n=\left\{1,2,\cdots,n\right\}\ and I_n be the partial one to one transformation semigroup on X_n under composition of mappings. If \left|x-y\right|=\left|x\alpha-y\alpha\right| \left(\forall x,y\in d o m\left(\alpha\right)\right) then \alpha is called length-preserving (distance-preserving or isometry) on X_n where \alpha\in I_n. Then the subset\ DP_n=\left\{{\alpha\in I}_n:\ \ \ \left|x-y\right|=\left|x\alpha-y\alpha\right|\left(\forall x,y\in d o m(\alpha)\right)\ \right\} is a subsemigroup of I_n. In this thesis we will compile some studies that contain important results about cardinalities, generating sets and ranks of DP_n and present them to the information of the researchers.