A simple Lie group is a type of mathematical structure that is both a group and a differentiable manifold. It is "simple" in the sense that it has no nontrivial normal subgroups, meaning it cannot be broken down into smaller, simpler groups. These groups play a crucial role in various areas of mathematics and theoretical physics, particularly in the study of symmetries. Examples of simple Lie groups include the special linear group SL(n, ℝ) and the special orthogonal group SO(n). These groups are important in understanding the symmetries of geometric objects and in the formulation of physical theories, such as quantum mechanics and general relativity.