# Equivalence of definitions of local powering-invariant subgroup

From Groupprops

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 of a group :

- Whenever and are such that there is a unique such that , we must have .
- Whenever and is a prime number such that there is a unique such that , we must have .

## Facts used

## Proof

### (1) implies (2)

This is obvious.

### (2) implies (1)

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