Equivalence of definitions of local powering-invariant subgroup

This article gives a proof/explanation of the equivalence of multiple definitions for the term local powering-invariant subgroup
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a subgroup $H$ of a group $G$:

1. Whenever $h\in H$ and $n\in \mathbb {N}$ are such that there is a unique $x\in G$ such that $x^{n}=h$, we must have $x\in H$.
2. Whenever $h\in H$ and $p$ is a prime number such that there is a unique $x\in G$ such that $x^{p}=h$, we must have $x\in H$.

Facts used

1. Unique nth root implies unique mth root for m dividing n

Proof

(1) implies (2)

This is obvious.

(2) implies (1)

We use Fact (1) to show uniqueness of $p^{th}$ root for some prime $p$ dividing $n$. Then, use condition (2) to obtain that that $p^{th}$ root is also in $H$. Now repeat the process with this new element -- we're now finding a $(n/p)^{th}$ root of the element. Each step gets rid of one prime divisor.