Equivalence of definitions of local powering-invariant subgroup

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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.