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

From Groupprops

## Statement

Suppose is a (finite or infinite) nilpotent group and is a normal subgroup of of order , where is a prime number. Then:

- is contained in the subgroup , where denotes the member of the Upper central series (?) of . Note that is a Characteristic subgroup (?) of of Nilpotency class (?) at most .
- In particular, the Characteristic closure (?) of in has nilpotency class at most .

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

## Related facts

### 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