Izometries of lie groups

This thesis consist of three chapters.In the firs chapter;fundamental definitions and theorems are given.In the second chapter is searched that finite groups, symetric groups and connections of this groups with Lie groups. After, it is studied that connected Lie groups, homotopy groups and fundamental group and apply Gauss Bonnet theorem. In this chapter latest it is studied that isometries of left invariant Riemann metric which is defined on Lie groups.The third chapter contain original part of our study. In this chapter it is studied that three dimensional Heisenberg group which is application range of Lie groups. It is given that properties of Lorentz metrics which are defined on three dimensional Heisenberg group and it is find out that equations of homogeneous geodesics in three dimensional Heisenberg group. In this chapter latest it is stuied that geodesically completeness of these Lorentz metrics metrics.