# Center is quotient-local powering-invariant in nilpotent group

From Groupprops

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., quotient-local powering-invariant subgroup)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

Suppose is a nilpotent group and is the center of . Then, is a quotient-local powering-invariant subgroup of : if any element of has a unique root in for some prime number , then its image in has a unique root in .

## Related facts

### Opposite facts

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

## Facts used

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

## Proof

The proof follows directly by combining Facts (1), (2), and (3).