Transformation groups in unitary spaces and the complete systems of invariants of points

In this thesis, the following main results have been obtained:1)Let H:C^n?R^2n be the natural linear operator from the n-dimensional C^n complex vector space to the 2n-dimensional R^2n real vector space. For an arbitrary linear transformation F:C^n?C^n, the relation between the determinants det(F) and det(HF(H^(-1))) is found.2)In the n-dimensional unitary space, an arbitrary isometry is a composition of a translation and a real unitary transformation.3)Let G be the unitary group or the group of all isometries of C^n. In the n-dimensional unitary space, complete systems of G-invariants have been obtained for finite subsets of C^n. The minimality properties of these complete systems of G-invariants have been proved.