| Summary: In this thesis, Equivalence Problems are investigated for ali isometries group of n-dimensional Euclidean spaces; Iz(n) and their important subgroups, group of orthogonal transformation; 0(n), group of translation and rotation; SIz(n), group of rotation SO(n) with systems of points in Rn. The equivalence problem has been solved for following groups Iz(n), 0(n), SIz(2), SO(2) with respect to invariant theory.Equivalence condition of Iz(n) in this thesis was found differently given equivalance condition in the book that the name is " Geometry I" of Berger.
Key Words: Euclidean Geometry, Isometry, Orthogonal Group, invariant.
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