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.