A Galois group is a mathematical concept that arises in the field of abstract algebra, specifically in the study of field extensions. It consists of a set of symmetries, or automorphisms, of a field extension that preserve the structure of the field. These symmetries help to understand the solutions of polynomial equations and their relationships. The significance of Galois groups lies in their ability to connect field theory and group theory. They provide insights into the solvability of polynomials by radicals, as established by Évariste Galois. By analyzing the Galois group of a polynomial, mathematicians can determine whether its roots can be expressed using simple arithmetic operations and radicals.