Use the fact that S5 is not solvable to prove that Sn is not

Use the fact that S5 is not solvable to prove that Sn is not solvable for n 5

Solution

S₅ is not solvable.

If S₅ were solvable, then so is A₅. But A₅ is simple and nonabelian, so not solvable. Hence S₅ is not solvable. Since S₅ is isomorphic to a subgroup of S_n for n > 5, S_n is not solvable for n >= 5.