# Homework Help: Group theory 2 problems

1. Jul 24, 2010

### losiu99

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

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
Lagrange theorem I think. Isomorphism theorems maybe?

3. 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.

2. Jul 25, 2010

### 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.