# Upper central series members are quotient-local powering-invariant in nilpotent group

From Groupprops

## Contents

## Statement

Suppose is a nilpotent group. Then, all members of the upper central series of are quotient-local powering-invariant subgroups and hence also local powering-invariant subgroups in .

## Related facts

### Similar facts

- Center is local powering-invariant
- Center is quotient-local powering-invariant in nilpotent group
- Upper central series members are local powering-invariant in Lie ring

### Opposite facts

- Second center not is local powering-invariant in solvable group
- Center not is quotient-local powering-invariant in solvable group

## Facts used

### Style (A)

- Center is quotient-local powering-invariant in nilpotent group
- Nilpotency is quotient-closed
- Quotient-local powering-invariance is quotient-transitive
- Quotient-local powering-invariant implies local powering-invariant

### Style (B)

- Center is local powering-invariant
- Local powering-invariance is quotient-transitive in nilpotent group
- Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group

## Proof

The proof method used in this article is discussed in the survey article inductive proof methods for the ascending series corresponding to a subgroup-defining function.|See a list of facts whose proof uses this method

### Using Style (A)

The proof of quotient-local powering-invariance follows directly from Facts (1)-(3), using the principle of mathematical induction. The proof of local powering-invariance follows by combining with Fact (4).

### Using Style (B)

This proof is also fairly straightforward from the given facts.