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Math Help - Groups of order 4 are abelian

  1. #1
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    Groups of order 4 are abelian

    Given |G| = 4, we are to show G is abelian (commutative).
    The proof wants me to start by assuming the existence of x in G s.t. o(a) = 4, and in that case, G is abelian. (Further cases follow.)

    How? All I can figure from that is that a = e.
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  2. #2
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    Re: Groups of order 4 are abelian

    Hey phys251.

    Could you show this by exhaustion (since you only have four elements)?
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  3. #3
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    Re: Groups of order 4 are abelian

    Hi,
    I assume from your question that you're just starting to learn about groups. Here's a rather long winded proof that groups of order 4 are abelian. (With a little more knowledge, you can prove that any group of order p2 for p a prime is abelian.)

    Groups of order 4 are abelian-mhfgroups15.png
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