# Powering-invariant over quotient-powering-invariant implies powering-invariant

## Statement

Suppose is a group and are subgroups of such that , is normal in , and the following conditions hold:

- is a quotient-powering-invariant subgroup of .
- is a powering-invariant subgroup of .

Then, is a powering-invariant subgroup of .

## Related facts

## Proof

**Given**: is a group and are subgroups of such that , is normal in , and the following conditions hold:

- is a quotient-powering-invariant subgroup of .
- is a powering-invariant subgroup of .

is a prime number such that is powered over . An element .

**To prove**: There exists such that .

**Proof**: Let be the quotient map.

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | There exists such that . | is -powered. | Given-direct. | ||

2 | is -powered. | is quotient-powering-invariant in , is -powered. | Given-direct. | ||

3 | and is the unique root of in . | Steps (1), (2) | Applying the homomorphism to Step (1) gives that . Since, by Step (2), is -powered, must be the unique root. | ||

4 | . | is powering-invariant in . | Steps (2), (3) | By Step (2), is -powered, hence is -powered from the given information. Thus, the unique root of in , which equals ,(from Step (3)) must be in . | |

5 | . | Step (4) | Since , , which is since . |