# Nilpotent implies intersection of normal subgroup with upper central series is strictly ascending till the subgroup is reached

From Groupprops

## Statement

Suppose is a nilpotent group and is a normal subgroup of . Suppose denotes the member of the upper central series of , i.e., is the trivial subgroup, is the center, and is the center of .

Let be the smallest nonnegative integer such that . Then is a proper subgroup of for all .

## Related facts

### Weaker facts

- Nilpotent implies center is normality-large: A special case, where we only use that for .

### Applications

- 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 .