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Math Help - Homomorphic images of D3.

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    Homomorphic images of D3.

    If Z2 is a homomorphic image of D3, what is the homomorphism?

    Φ: D3 to Z2
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  2. #2
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    Quote Originally Posted by o&apartyrock View Post

    If Z2 is a homomorphic image of D3, what is the homomorphism?

    Φ: D3 to Z2
    we have D_3=<a,b: \ a^2=b^3=1, \ ab=b^{-1}a>=\{1,a,b,b^2,ab,ab^2 \}. let f: D_3 \to \mathbb{Z}/2 be an onto homomorphism. then we must have |\ker f|=3.

    the only normal subgroup of D_3 of order 3 is <b>. so we must have f(1)=f(b)=f(b^2)=\bar{0}, \ f(a)=f(ab)=f(ab^2)=\bar{1}.
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