Prime graphs of Finite Groups

All the groups treated in the thesis are finite. Let X be a set of integers and consider the primes that divide some element of X as vertices, where two vertices are adjacent if their product divides some element of X. The graph obtained is called the prime graph on X. In this thesis we give an algebraic description of a group G in the following three cases. 1. When the Gruenberg-Kegel graph of G (that is the prime graph on the set of all group element orders) has a cut-set. 2. When the prime graph on the degrees of irreducible real characters of G has no edges. 3. When the prime graph on the lengths of real conjugacy classes has no edges. Is presented also a description of groups acting on a module in the case that the lengths of orbits are prime powers.