Nilpotent implies intersection of normal subgroup with upper central series is strictly ascending till the subgroup is reached
Let be the smallest nonnegative integer such that . Then is a proper subgroup of for all .
- Nilpotent implies center is normality-large: A special case, where we only use that for .
- Normal of prime power order implies contained in upper central series member corresponding to prime-base logarithm of order in nilpotent: This says that a normal subgroup of order in a nilpotent group is contained in .