Is there any connection between Evariste Galois's group theory and symmetry? - Answers

There sure is, and a major connection at that.Consider a finite set of n elements. The symmetric group of this set is said to have a degree of n. The symmetric group of degree n (Sn) is the Galois group of the general polynomial of degree n. In order for there to be a formula involving radicals that solve the general polynomial of degree n, such as the quadratic equation when n = 2, that polynomial's corresponding Galois group must be solvable. S5 is not a solvable group. Therefore, the Galois group of the general polynomial of degree 5 is not solvable. Thus the general polynomial of degree 5 has no general formula to solve it using radicals.This was huge result, and one of the first real applications, for group theory, since that problem had stumped mathematicians for centuries.