Normal subgroup with prime index

[oyolasigmaz]

Homework Statement

Prove that if p is a prime and G is a group of order p^a for some a in Z+, then every subgroup of index p is normal in G.

Homework Equations

We know the order of H is p^(a-1). H is a maximal subgroup, if that matters.

The Attempt at a Solution

Suppose H≤G and (G)=p but H is not a normal subgroup of G. So for some g in G Hg≠gH. I know I didn't do much, but is this the correct way to start? What to do now?

 

[jbunniii]

Do you know any general theorems about normalizers in p-groups?

 

[oyolasigmaz]

I am not sure about which theorem you are talking about, but I just found a theorem giving me the result I want in Dummit and Foote stating that if n is the order of the group and p the largest prime dividing n, then I have the result I wanted.

 

[jbunniii]

The theorem I was talking about is that "normalizers grow" in p-groups. This means that if G is a p-group and H < G is a proper subgroup, then $H < N_G(H)$, i.e. H is a proper subgroup of its normalizer. (This is in fact true if G is any finite nilpotent group.)

Therefore if H < G and H has index p, $N_G(H)$ must be all of G.