Transitivity is not disjunction-closed

This article gives the statement, and possibly proof, of a subgroup metaproperty not satisfying a subgroup metametaproperty .
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Statement

First unpacking

It is possible to have two transitive subgroup properties $p$ and $q$ such that the disjunction (OR) of $p$ and $q$ is not a transitive subgroup property.

Second unpacking

It is possible to have two transitive subgroup properties $p$ and $q$ and groups $H\leq K\leq G$ such that $H$ satisfies $p$ or $q$ in $K$, and $K$ satisfies $p$ or $q$ in $G$, but $H$ satisfies neither $p$ nor $q$ in $G$.

Proof

The construction of the proof is as follows:

• $p$: The property of a subgroup having index a power of 2 (including the possibility of the index being 1)
• $q$: The property of a subgroup having index a power of 2 (including the possibility of the index being 1)
• $G$: cyclic group:Z6
• $H$: trivial subgroup of $G$
• $K$: cyclic subgroup of order 3 in $G$

With this construction, $H$ satisfies $q$ in $K$ and $K$ satisfies $p$ in $G$. $H$ satisfies neither $p$ nor $q$ in $G$.