TITLE: Number of Normal subgroups In a p-Group
QUESTION [8 upvotes]: Dear all,
Does someone know of any paper/method that enables us counting/estimating the number of normal subgroups of some p-group of order $p ^n $ ($ n$ is some natural number ? ) .
Is there anyway we can count the maximal subgroups it has (i.e.- the groups of order $p^{n-1} $ ? ) ?
Thanks in advance

REPLY [4 votes]: The wikipedia article on p-groups reminded me that

Every normal subgroup of a finite
  p-group intersects the center
  nontrivially.

This implies immediately that minimal normal subgroups of a p-group $G$ will be central. This fact can be used to prove the statement that Wei Zhou made:

A $p$-group of maximal class and size $p^n$ has the least number of normal subgroups of all groups of order $p^n$.

(If I'm thinking straight this number is $n+1$ and the bound is also achieved by the cyclic group of order $p^n$.)
It seems to me that one might be able to prove something a little stronger using an inductive argument: counting the minimal normal subgroups in the center $Z$, and then counting the normal subgroups in $G/Z$, and then putting these two numbers together... It's that last bit that's going to be tricky though. If the center is cyclic, then everything is fine but when it's not cyclic, eek...