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My Proof of Structure Theorem for Finite Abelian Groups

  1. Mar 17, 2013 #1
    Hello! If anybody has a minute, I'd appreciate a quick look-through of my proof that a finite abelian group can be decomposed into a direct product of cyclic subgroups. I'm new to formal writing (as well as Latex) and all feedback is greatly appreciated!

    Thanks in advance for your time!

    http://www.scribd.com/doc/130897466/Structure-of-Finite-Abelian-Groups-Brian-Blake
     
  2. jcsd
  3. Mar 18, 2013 #2

    jbunniii

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    Your Lemma 1 is not valid. Consider for example a cyclic group ##G = \langle x\rangle## with ##|G| = p^2##. The subgroup ##H = \langle x^p \rangle## has order ##p##, so ##G/H## and ##H## are both cyclic with order ##p##. But clearly ##G/H \times H## is not cyclic; it has order ##p^2## but any element has order ##1## or ##p##.

    The problem is your claim that ##\phi(g_1 g_2) = (x^{k+j}H, h_1 h_2)##. This is not true in general. Let's take a concrete example with ##G## as above.

    Suppose ##p = 3##, so ##G = \{e, x, x^2, \ldots x^8\}##, and ##H = \{e, x^3, x^6\}##, and ##G/H = \{H, xH, x^2 H\}##.

    Take ##g_1 = x^{4} = x^1 x^3 \in x H## and ##g_2 = x^{5} = x^2 x^3 \in x^2 H##. Then ##g_1 g_2 = x^{9} = e = x^0 x^0 \in H##.

    Then:
    $$\phi(g_1) = (x H, x^{3})$$
    $$\phi(g_2) = (x^2 H, x^{3})$$
    but
    $$\phi(g_1 g_2) = (H, x^{0})$$
    whereas
    $$\phi(g_1)\phi(g_2) = (x H \cdot x^2 H, x^{3}\cdot x^{3}) = (H, x^{6}) \neq \phi(g_1 g_2)$$
     
    Last edited: Mar 18, 2013
  4. Mar 18, 2013 #3

    jbunniii

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    Sorry, I had to make a few minor edits. Please refresh if you've already read the post.
     
  5. Mar 18, 2013 #4
    Thank you very much! I guess I was thinking incorrectly that the order of x was also the order of the quotient group. Back to the drawing board!
     
  6. Mar 18, 2013 #5

    jbunniii

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    No problem, feel free to post an update when you have a revised proof. It was a good attempt - I knew the conclusion of Lemma 1 was wrong, but it took me a while to spot the problem with your proof.

    Also, your writing style and Latex are good, so nothing to worry about there.
     
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