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Is this true?

H ⊴ K and K ⊴ G imply H ⊴ G

No — it is false.

The counterexample

⟨(12)(34)⟩ ⊴ ⊴

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

The kind of mistake this is

Property not inherited

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

Drill statements like this
groups

Others that fail the same way

From Groups › Normal subgroups, quotients, isomorphism theorems

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