1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Group theory 2 problems

  1. Jul 24, 2010 #1
    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. jcsd
  3. Jul 25, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook