This paper discusses the contraction of groups and their representations, focusing on how groups can be limiting cases of other groups and how their representations can be derived from those of the groups they are limits of. The contraction process involves transforming the infinitesimal elements of a group through a singular transformation, leading to a new group. The contraction of the inhomogeneous Lorentz group with respect to the subgroup of time displacements yields the homogeneous Galilei group, while contraction of the de Sitter groups yields the inhomogeneous Lorentz group. The contraction of the three-dimensional rotation group results in the Euclidean group for two dimensions. The paper also explores the contraction of representations, showing how representations of the contracted group can be obtained from those of the original group. For example, the contraction of the inhomogeneous Lorentz group with respect to the subgroup of spatial rotations and time displacements yields the full Galilei group. The contraction of the unitary representations of the Lorentz groups leads to the Galilei group, which is the group of classical mechanics. The paper discusses the structure of these representations and their convergence as the contraction parameter approaches zero. It also addresses the physical significance of these representations, noting that those corresponding to imaginary masses are not physically meaningful. The paper concludes with the contraction of the Klein-Gordon equation representation, showing how it can be transformed into the Schrödinger equation representation through a suitable contraction process.This paper discusses the contraction of groups and their representations, focusing on how groups can be limiting cases of other groups and how their representations can be derived from those of the groups they are limits of. The contraction process involves transforming the infinitesimal elements of a group through a singular transformation, leading to a new group. The contraction of the inhomogeneous Lorentz group with respect to the subgroup of time displacements yields the homogeneous Galilei group, while contraction of the de Sitter groups yields the inhomogeneous Lorentz group. The contraction of the three-dimensional rotation group results in the Euclidean group for two dimensions. The paper also explores the contraction of representations, showing how representations of the contracted group can be obtained from those of the original group. For example, the contraction of the inhomogeneous Lorentz group with respect to the subgroup of spatial rotations and time displacements yields the full Galilei group. The contraction of the unitary representations of the Lorentz groups leads to the Galilei group, which is the group of classical mechanics. The paper discusses the structure of these representations and their convergence as the contraction parameter approaches zero. It also addresses the physical significance of these representations, noting that those corresponding to imaginary masses are not physically meaningful. The paper concludes with the contraction of the Klein-Gordon equation representation, showing how it can be transformed into the Schrödinger equation representation through a suitable contraction process.