Pure definability is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., purely definable subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about purely definable subgroup |Get facts that use property satisfaction of purely definable subgroup | Get facts that use property satisfaction of purely definable subgroup|Get more facts about quotient-transitive subgroup property

Statement

Statement with symbols

Suppose $H\leq K\leq G$ are groups such that $H$ is a purely definable subgroup of $G$ and $K/H$ is a purely definable subgroup of $G/H$ (note that any purely definable subgroup is characteristic and hence normal, so it makes sense to take the quotient group). Then, $K$ is a purely definable subgroup of $G$.