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Math Help - Prove that G is abelian iff...

  1. #1
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    Prove that G is abelian iff...

    Prove that G is abelian if and only if the map  f: G \longrightarrow G given by  f(g)=g^2 is a group homomorphism.

    f(g_{1} \cdot g_{2})=g_{1}g_{2}g_{1}g_{2}=g_{1}g_{1}g_{2}g_{2}=g  _{1}^2g_{2}^2

    if G abelian then f is homomorphism.
    Nevertheless i prove only one direction of the statement. Can anybody help me?
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  2. #2
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    Quote Originally Posted by Magics6 View Post
    Prove that G is abelian if and only if the map  f: G \longrightarrow G given by  f(g)=g^2 is a group homomorphism.

    f(g_{1} \cdot g_{2})=g_{1}g_{2}g_{1}g_{2}=g_{1}g_{1}g_{2}g_{2}=g  _{1}^2g_{2}^2

    if G abelian then f is homomorphism.
    Nevertheless i prove only one direction of the statement. Can anybody help me?

    If f is a homom. then for any a\,,\,b\in\,G\,,\,\,abab=aabb . Now cancel stuff here and show abelianess.

    Tonio
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