Is the following statement true? H ⊴ K and K ⊴ G imply H ⊴ G

No. ⟨(12)(34)⟩ ⊴ V₄ ⊴ A₄ Conjugating (12)(34) by the 3-cycle (123) gives (13)(24) ∉ ⟨(12)(34)⟩.

Counterexample

Is this true?

H ⊴ K and K ⊴ G imply H ⊴ G

No — it is false.

The counterexample

⟨(12)(34)⟩ ⊴ $V_{4}$ ⊴ $A_{4}$

Why it fails

Conjugating (12)(34) by the 3-cycle (123) gives (13)(24) ∉ ⟨(12)(34)⟩.

Keep practicing

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.