Upper central series members are quotient-powering-invariant

Statement
Suppose $$G$$ is a group. Denote by $$Z^{(0)}(G),Z^{(1)}(G),Z^{(2)}(G),\dots$$ the members of the upper central series of $$G$$, all of which are subgroups of $$G$$. Here, $$Z^{(0)}(G)$$ is the trivial subgroup, $$Z^{(1)}(G)$$ is the center, $$Z^{(2)}(G)$$ is the second center, and so on. Then, all the subgroups $$Z^{(i)}(G)$$, for nonnegative integers $$i$$, are quotient-powering-invariant subgroups of $$G$$. In particular, since quotient-powering-invariant implies powering-invariant, they are all powering-invariant subgroups of $$G$$.

The result can in fact be extended to the transfinite upper central series.

Related facts

 * Lower central series members are quotient-powering-invariant in nilpotent group
 * Normal subgroup contained in the hypercenter that is powering-invariant is quotient-powering-invariant

Facts used

 * 1) uses::Center is quotient-powering-invariant
 * 2) uses::Quotient-powering-invariance is quotient-transitive
 * 3) uses::Quotient-powering-invariance is union-closed

Proof for the finite part
The proof follows directly by combining Facts (1) and (2), and using the principle of mathematical induction (just a single application would get us to the second center).

Proof for the transfinite upper central series
This basically follows the same way, but we need to use transfinite induction instead. The additional ingredient we need is the argument for limit ordinals, and this basically follows from Fact (3).

Journal references

 * , Corollary 13.4, Page 230 (14th page in the paper)