# Omega-1 of center is normality-large in nilpotent p-group

From Groupprops

## Statement

Let be a nilpotent p-group, i.e., a nilpotent group where the order of every element is a power of the prime . Then, the subgroup is a normality-large subgroup of : its intersection with every nontrivial normal subgroup is nontrivial.

Here, denotes the omega subgroup: the subgroup generated by all the elements of order , and denotes the center of .

Note that if is a finite p-group, i.e., a group of prime power order, then it is nilpotent.

## Related facts

### Corollaries

- Minimal normal implies contained in Omega-1 of center for nilpotent p-group
- Socle equals Omega-1 of center for nilpotent p-group
- Minimal characteristic implies contained in Omega-1 of center for nilpotent p-group

## Facts used

- Nilpotent implies center is normality-large
- Omega-1 is large (and hence, is normality-large)
- Normality-largeness is transitive