Group Solvability Explorer — S₄ vs S₅
Commutator Reduction
Cayley Graph
Galois Connection
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Current group——
Derived subgroup——
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Reduction computes the commutator subgroup: G ← [G,G].
For S₄: S₄ → A₄ → V₄ → {e}.
For S₅: S₅ → A₅ → A₅ (stuck).
For S₄: S₄ → A₄ → V₄ → {e}.
For S₅: S₅ → A₅ → A₅ (stuck).
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Permutation displayed as [p(1) p(2) … p(n)]. Token i moves to position p(i).
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Generators—
Identity node is green. Pink/teal edges correspond to the two generators.
A₅ (60 nodes) is dense; hover nodes for the permutation tooltip.
The Fundamental Connection
A polynomial is solvable by radicals iff its Galois group is solvable.
Problem: solve x⁵ − x − 1 = 0 using only radicals.
S₄ (degree 4): S₄ → A₄ → V₄ → {e} ✓
S₅ (degree 5): S₅ → A₅ → A₅ → … ✗
S₅ (degree 5): S₅ → A₅ → A₅ → … ✗
Abel–Ruffini: since S₅ is not solvable, there is no general radical formula for quintics.