# Local powering-invariant subgroup containing the center is intermediately local powering-invariant in nilpotent group

From Groupprops

## Contents

## Statement

Suppose is a nilpotent group and is a subgroup containing the center of that is also a local powering-invariant subgroup of . Then, is an intermediately local powering-invariant subgroup of . Explicitly, suppose is a subgroup of containing . Then, is a local powering-invariant subgroup of .

## Related facts

## Facts used

- Torsion-freeness for a prime is subgroup-closed
- Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
- Nilpotency is subgroup-closed

## Proof

**Given**: A nilpotent group , a subgroup of that is local powering-invariant and such that where is the center of . A subgroup of containing . A prime number and an element such that there is a unique element satisfying .

**To prove**: There exists a unique element such that .

**Proof**:

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

1 | . | . | given-direct | ||

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

3 | is -torsion-free. | Fact (2) | has a unique root in . | We use the equivalence (3) implies (1) within the multi-part equivalence of Fact (2). | |

4 | is -torsion-free. | Fact (1) | Steps (1), (3) | Step-fact combination direct | |

5 | The map is injective in . | Fact (2) | is nilpotent | Step (4) | Step-fact combination direct (specifically, we want to use the implication from (4) to (1) in the multi-part equivalence of Fact (2)) |

6 | The element is the unique root of in all of . | satisfies . | Step (5) | given-step direct | |

7 | The element of Step (6) is in . | is local powering-invariant in . | Step (6) | Step-given combination direct. |