Powering-invariance is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Suppose $H\leq K\leq G$ are groups such that $H$ is a powering-invariant subgroup of $K$ and $K$ is a powering-invariant subgroup of $G$. Then, $H$ is a powering-invariant subgroup of $G$.

Related facts

• Divisibility-closedness is transitive
• Powering-invariance is not quotient-transitive
• Local powering-invariance is transitive

Facts used

1. Balanced implies transitive

Proof

Abstract proof

Powering-invariance can be expressed as the balanced subgroup property with respect to being a rational power map; in function restriction expression it can be written as:

Rational power map $\to$ Rational power map

In other words, a subgroup is powering-invariant if and only if any rational power map of the whole group restricts to a rational power map of the subgroup.

Transitivity now follows from fact (1).

NOTE: Instead of rational power map, we could use "prime root map" ($p^{th}$ roots for all primes $p$ for which such roots are unique in the group), and the proof would still hold. We could also use "integer root map" ($n^{th}$ roots for all integers $n$ for which such roots are unique in the group), and the proof would still hold.