# Powering is central extension-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., powered group for a set of primes) satisfying a group metaproperty (i.e., central extension-closed group property)

View all group metaproperty satisfactions | View all group metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about powered group for a set of primes |Get facts that use property satisfaction of powered group for a set of primes | Get facts that use property satisfaction of powered group for a set of primes|Get more facts about central extension-closed group property

## Contents

## Statement

Suppose is a group, is a prime number, and is a central subgroup of such that the following are true:

- is powered over .
- is powered over .

Then, is also powered over .

## Related facts

## Facts used

## Proof

Fact (1) gives the existence of roots in , whereas Fact (2) gives their uniqueness.