Determine whether your finite group is solvable here. Choose families, permutation generators, or Cayley tables. Get clear steps, series length, and downloads quickly now.
A group G is solvable if its derived series reaches the trivial subgroup. Define G^(0)=G and recursively G^(i+1) = [G^(i), G^(i)], where [H,H] is the subgroup generated by all commutators [a,b]=a^{-1}b^{-1}ab with a,b ∈ H.
The smallest k such that G^(k) = {e} is the derived length. If the series stabilizes at a nontrivial subgroup, the group is not solvable.
Example Cayley table for the cyclic group of order 4 under addition modulo 4 (labels 0..3).
| + | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 | 3 | 0 | 1 | 2 |
| Input | Example | Expected verdict |
|---|---|---|
| Known family | S₃ | Solvable |
| Known family | A₅ | Not solvable |
| Permutation generators | (1 2 3) | Solvable |
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.