Pure definability is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., purely definable subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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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 transitive subgroup property

Statement

Verbal statement

A purely definable subgroup of a purely definable subgroup is purely definable.

Statement with symbols

Suppose $H\leq K\leq G$ are groups such that $K$ is a purely definable subgroup of $G$ and $H$ is a purely definable subgroup of $K$. Then, $H$ is a purely definable subgroup of $G$.

Related facts

Similar facts

• Pure definability is quotient-transitive
• Second-order definability is transitive
• Second-order characteristicity is transitive
• Monadic second-order definability is transitive
• Monadic second-order characteristicity is transitive