Normal of prime power order implies contained in upper central series member corresponding to prime-base logarithm of order in nilpotent

Statement
Suppose $$G$$ is a (finite or infinite) nilpotent group and $$H$$ is a normal subgroup of $$G$$ of order $$p^r$$, where $$p$$ is a prime number. Then:


 * 1) $$H$$ is contained in the subgroup $$Z^{r}(G)$$, where $$Z^r$$ denotes the $$r^{th}$$ member of the fact about::upper central series of $$G$$. Note that $$Z^r(G)$$ is a fact about::characteristic subgroup of $$G$$ of fact about::nilpotency class at most $$r$$.
 * 2) In particular, the fact about::characteristic closure of $$H$$ in $$G$$ has nilpotency class at most $$r$$.

The typical application of this is when $$G$$ is itself a group of prime power order, i.e., the order is of the form $$p^n$$. Since prime power order implies nilpotent, the result always applies in this case.

Similar facts

 * Minimal normal implies central in nilpotent (which uses nilpotent implies center is normality-large)
 * Socle equals Omega-1 of center in nilpotent p-group
 * Normal of order equal to least prime divisor of group order implies central

Applications

 * Normal closure of 2-subnormal subgroup of prime power order in nilpotent group has nilpotency class at most equal to prime-base logarithm of order

Facts used

 * 1) uses::Nilpotent implies intersection of normal subgroup with upper central series is strictly ascending till the subgroup is reached