Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Isomorphisms between cyclic groups? (stupid question)

  1. Apr 19, 2009 #1
    Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?
  2. jcsd
  3. Apr 20, 2009 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    On what level do you want the 'why' explained? I suppose the 'real' answer is because 2 and 3 are coprime, but 2 is not coprime to 2.

    This is either a consequence of the structure theorem for abelian groups, or an explanation of that theorem, depending on how you look at it.

    We could look a bit deeper: suppose that G and H are cyclic, when is GxH cyclic? Well, suppose that we claim (g,h) is a cyclic generator of GxH. Well, g must be a generator of G with order p, say, and h a generator of H with order q, and (g,h) must have order pq. If I raise (g,h) to the power p, then it is (1,k) for some k in H, and in order for (g,h) to have order pq k must have order q, i.e. also be a generator for H.

    Pulling out the important thing: H must have an element h of order so that h^p still has order q. But this is if and only if p is coprime to q (this is an elementary fact you may have learnt already).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook