Center is quotient-powering-invariant

Statement
Suppose $$G$$ is a group and $$Z(G)$$ denotes the center of $$G$$. Then, $$Z(G)$$ is a quotient-powering-invariant subgroup of $$G$$. Explicitly, if $$p$$ is a prime number such that $$G$$ is powered over $$p$$, the quotient group $$G/Z(G)$$ (which can be identified with the inner automorphism group of $$G$$) is also powered over $$p$$.

Facts used
We essentially use that the center is a fixed-point subgroup of a subgroup of the automorphism group (for Fact (1)) and that it is a central subgroup (for Fact (2)) to get the result.


 * 1) uses::Center is local powering-invariant (and hence, the center is powering-invariant). This is a special case of the fact that fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
 * 2) uses::Powering-invariant and central implies quotient-powering-invariant

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