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Math Help - Homomorphism

  1. #1
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    Homomorphism

    How can I show that there is no homomorphism from S_3 to Z_3?
    Suppose there is one homomorphism f. Let g \in S_3, then f(g)=0 or f(g)=1 or f(g)=2. I know ord(f(g)) must divide ord(g). I have ord(0)=\infty, ord(1)=3, and ord(2)=2. I want to show that there is an element in S_3 such that ord(f(g)) does not divide ord(g). I'm stuck on this because based on my approach, ord(f(g)) can be 1, and 1 divides anything.

    Can anyone give me help here?
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  2. #2
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    Quote Originally Posted by jackie View Post
    How can I show that there is no homomorphism from S_3 to Z_3?
    Suppose there is one homomorphism f. Let g \in S_3, then f(g)=0 or f(g)=1 or f(g)=2. I know ord(f(g)) must divide ord(g). I have ord(0)=\infty, ord(1)=3, and ord(2)=2. I want to show that there is an element in S_3 such that ord(f(g)) does not divide ord(g). I'm stuck on this because based on my approach, ord(f(g)) can be 1, and 1 divides anything.

    Can anyone give me help here?

    Who says there is no homom.? There is always the trivial homom.: send everything to the unity element. But for this there is no more since:

    If there was a non-trivial hom. F: S_3 --> Z_3 it would have to be onto ==> Then S_3/Ker F ~ Z_3, but this'd mean S_3 has a normal sbgp. (namely, ker F) of order 2, which isn't true.

    Tonio
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