Abel-Ruffini and solvable groups: why S_5 blocks the quintic

Computes the derived series of the symmetric groups S_2 to S_5. It reaches the trivial group for n <= 4 but stalls at A_5 for n = 5, the group-theoretic reason the general quintic has no formula in radicals.

from mathematicskit.abstract_algebra import PermutationGroup, derived_series, is_solvable

Derived series of S_n

for n in range(2, 6):
    group = PermutationGroup(n)
    orders = [len(h) for h in derived_series(group)]
    print(f"S_{n}: derived series orders {orders} -> solvable: {is_solvable(group)}")

S_2: derived series orders [2, 1] -> solvable: True
S_3: derived series orders [6, 3, 1] -> solvable: True
S_4: derived series orders [24, 12, 4, 1] -> solvable: True
S_5: derived series orders [120, 60] -> solvable: False

The obstruction

The series for S_5 stops at A_5 (order 60), which equals its own commutator subgroup. By Galois’s criterion, a degree-5 polynomial whose Galois group is S_5 cannot be solved by radicals.