# Powering-injectivity is inherited by extensions where the normal subgroup is contained in the hypercenter

From Groupprops

## Statement

Suppose we have the following:

- A group .
- A normal subgroup of .
- A set of prime numbers.

such that the following are satisfied:

- is a normal subgroup contained in the hypercenter in (note that this condition is automatically satisfied if is a nilpotent group).
- Both and are -powering-injective groups. Note that for , this is equivalent to being -torsion-free.

Then, is also a -powering-injective group.

## Facts used

- Powering-injectivity is inherited by central extensions
- Something to show that each of the groups is -torsion-free, similar to equivalence of definitions of nilpotent group that is torsion-free for a set of primes.