Metabelian group
A group G is metabelian if its derived subgroup [G, G] is abelian.
Equivalent conditions:
- derived length ≤ 2 (the derived series reaches {e} in ≤ 2 steps: G ▷ [G,G] ▷ {e}).
- there exists an abelian normal subgroup N ⊴ G with G/N abelian.
Key facts
- Every abelian group is metabelian (derived length ≤ 1).
- Metabelian ⟹ solvable.
- The class is closed under subgroups and quotients.
Examples
S₃,D₄,Q₈— each has abelian derived subgroup.A₄: [A₄,A₄] = V₄ = C₂×C₂ is abelian, and A₄/V₄ ≅ C₃.- Every group of order p², pq, or p³.
- Every dihedral and dicyclic group.
Non-examples
S₄: derived series S₄ ▷ A₄ ▷ V₄ ▷ {e} has length 3, so [S₄,S₄] = A₄ is not abelian.- S₅ likewise (derived subgroup A₅ is nonabelian simple).