# Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group

From Groupprops

## Contents

## Statement

Suppose is a nilpotent group and is a subgroup of . The following are equivalent:

- is a local powering-invariant normal subgroup of , i.e., is both a local powering-invariant subgroup of and a normal subgroup of .
- is a quotient-local powering-invariant subgroup of .

## Related facts

### Opposite facts

- Local powering-invariant and normal not implies quotient-local powering-invariant
- Center not is quotient-local powering-invariant

### Applications

- Local powering-invariance is quotient-transitive in nilpotent group
- Upper central series members are local powering-invariant in nilpotent group

## Facts used

- Quotient-local powering-invariant implies local powering-invariant
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
- Nilpotency is quotient-closed

## Proof

### (2) implies (1)

This follows from Fact (1) (note that normality is definitional).

### (1) implies (2)

**Given**: A nilpotent group , a subgroup of . is normal and local powering-invariant in . An element and a prime number such that the equation has a unique solution . is the quotient map.

**To prove**: is the unique root of in the quotient group .

**Proof**:

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

1 | The map is injective from to itself. | Fact (2) (implication (3) implies (1) in the equivalence on the fact page) | is nilpotent and has a unique root. | Step-fact combination direct. | |

2 | For any satisfying , we have . | is local powering-invariant in . | Step (1) | Let . By Step (1), the -power map is injective in , hence is the unique root of in . Since is local powering-invariant, this forces . | |

3 | is -torsion-free. | Step (2) | If an element of has order , any inverse image of it in is an element outside whose power is in , contradicting Step (2). | ||

4 | is nilpotent. | Fact (3) | is nilpotent | Given-fact direct | |

5 | The map is injective from to itself. | Fact (2) | Steps (3), (4) | Direct by combining the steps and the fact. | |

6 | is the unique root of in . | Step (5) | Applying to gives that . The injectivity of the map by Step (5) now gives uniqueness. |