What are the properties of normal subgroups in groups of prime order?

[losiu99]

Hi, I've been vanquished by probably easy problems once again.

Homework Statement

1. Let G be a group of order p^2 (p prime number), and H its subgroup of order p. Show that H is normal. Prove G must be abelian.

2. If a group G has exactly one subgroup H of order k, prove H is normal in G.

Homework Equations

Lagrange theorem I think. Isomorphism theorems maybe?

The Attempt at a Solution

1. Obviously H is cyclic. If H is not normal, G cannot be abelian, hence all the elements are of order p, except for the neutral one. G is therefore divided into p+1 cyclic, disjoint (except for e) subgroups of order p. So far I haven't succeeded deriving a contradiction.

2. Normalizer is either H or the whole group. Perhaps some property of self-normalizing groups yields a contradiction?

Thank you very much for any hints.

 

[Dick]

The second one isn't hard at all. Just think about the conjugate subgroups of H. The first one is a little harder. I'd start by using the class equation to show that G has a nontrivial center.