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Math Help - abelian group

  1. #1
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    abelian group

    Hi!

    I've the next problem:

    Be G a group with neuter e. Proof the next affirmation:

    If a^2 = e\, \forall a\in G\, \Rightarrow \, G is abelian

    I tried to proof the equivalent:

    If G is not abelian \Rightarrow\, a^2\neq e

    Proof:
    I suppose that a^2 = e
    a^2 = a . a = a . a^{-1} = a^{-1} . a = e\, \mbox{ ABSURDUM G is not abelian }

    This proof not convinced me so I ask for your help.

    Thank's
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Jagger View Post
    Hi!

    I've the next problem:

    Be G a group with neuter e. Proof the next affirmation:

    If a^2 = e\, \forall a\in G\, \Rightarrow \, G is abelian

    I tried to proof the equivalent:

    If G is not abelian \Rightarrow\, a^2\neq e

    Proof:
    I suppose that a^2 = e
    a^2 = a . a = a . a^{-1} = a^{-1} . a = e\, \mbox{ ABSURDUM G is not abelian }

    This proof not convinced me so I ask for your help.

    Thank's
    The negation would be that if G is no abelian there is some element of order greater than two.

    All you need to note is that ab=a(ab)^2b=a^2bab^2=ba
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