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