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Math Help - group homomorphism

  1. #1
    nhk
    nhk is offline
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    group homomorphism

    24. Suppose that ƒ:Z50 to Z15 (co-domain is Z 15) is a group homomorphism with ƒ(7)=6.
    a. Determine ƒ(x).
    b. Determine ƒ(Z50).
    c. Determine ker ƒ.
    d. Determine ƒ–1({3}).


    I am lost particularly on b, a,c and d I think I will be able to get by myself. Any suggestions to help me for b?
    Thanks
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  2. #2
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    Quote Originally Posted by nhk View Post
    24. Suppose that ƒ:Z50 to Z15 (co-domain is Z 15) is a group homomorphism with ƒ(7)=6.
    a. Determine ƒ(x).
    b. Determine ƒ(Z50).
    c. Determine ker ƒ.
    d. Determine ƒ–1({3}).


    I am lost particularly on b, a,c and d I think I will be able to get by myself. Any suggestions to help me for b?
    Thanks

    First, as (7,50)=1 then <7>=\mathbb{Z}_{50} and then \forall\,x\in\mathbb{Z}_{50}\,\,\exists\,n\in\math  bb{Z}\,\,\,s.t.\,\,\,7n=x \Longrightarrow f(x)=f(7n)=nf(7)=6n\!\!\!\pmod{15}.

    Since ord(6)=5\,\,\,in\,\,\,\mathbb{Z}_{15} , you get that f\left(\mathbb{Z}_{50}\right)=<6>\cong \mathbb{Z}_5 .

    I did two, now you do the other two.

    Tonio
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