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