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. Center is quotient-powering-invariant
2. Quotient-powering-invariance is quotient-transitive
3. Quotient-powering-invariance is union-closed

Proof

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).

References

Journal references

• Some aspects of groups with unique roots by Gilbert Baumslag, Acta mathematica, Volume 104, Page 217 - 303(Year 1960): ^{PDF (ungated)More info}, Corollary 13.4, Page 230 (14th page in the paper)