# Formula for number of minimal normal subgroups of group of prime power order

From Groupprops

## Statement

Suppose is a group of prime power order where the underlying prime is . Suppose is the minimum size of generating set for the center . Then, the number of minimal normal subgroups of equals:

All of these subgroups are of order and are contained in the center.

In particular, the number of such subgroups is congruent to 1 mod .

## Facts used

- Minimal normal implies contained in Omega-1 of center for nilpotent p-group (which follows from nilpotent implies center is normality-large)
- Central implies normal
- Equivalence of definitions of size of projective space

## Proof

By facts (1) and (2), the minimal normal subgroups are precisely the subgroups of order contained inside (the subgroup generated by elements of order in the center), which is an elementary abelian group of order . Fact (3) now tells us that the number of such subgroups is given by the indicated formula.