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Thread: A group homomorphism that doesn't send one into one.

  1. #1
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    A group homomorphism that doesn't send one into one.

    Let $G$ be a group of order 3, $G= <g>$ and let $H$ be a group with an element h of order 4. Let $f: G --> H$, where $f(g^i)= h^i$. Then f is not an homomorphism because $f(g^3)= h^3$ but $g^3= 1$ and $h^3 \neq 1$. However $f(g^ig^j)= f(g^{i+j})= h^{i+j}= h^i h^j= f(g^i)f(g^j)$. Where is the mistake?

    In particular $f(gg^2)=f(g^3)=h^3=hh^2=f(g)f(g^2)$ and again the contradiction.
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  2. #2
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    Re: A group homomorphism that doesn't send one into one.

    $\displaystyle f:G \longrightarrow H$

    $f\left(g^i\right)=h^i$ for all $i$

    is not a well defined function
    Thanks from topsquark
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  3. #3
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    Re: A group homomorphism that doesn't send one into one.

    You are right. $g^4=g$ but $f(g^4) \neq f(g)$. Thanks a lot.
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