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Math Help - homomorphism again.

  1. #1
    Junior Member
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    Talking homomorphism again.

    Determine homomorphism f:\mathbb{Z}\rightarrow\mathbb{Z}_7 such that f(1)=4.
    And then determine ker(f)\text{ and }f(25).
    Last edited by mr fantastic; November 16th 2009 at 02:31 AM. Reason: Edited post title
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  2. #2
    Senior Member Sampras's Avatar
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    Quote Originally Posted by GTK X Hunter View Post
    Determine homomorphism f:\mathbb{Z}\rightarrow\mathbb{Z}_7 such that f(1)=4.
    And then determine ker(f)\text{ and }f(25).
    That is a fixed point.
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  3. #3
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    Since f is a homomorphism, f(x+ y)= f(x)+ f(y) for all x and y in Z and from that, by induction on n, f(nx)= n f(x) for all x in Z and n a positive integer. Now just calculate:
    f(25)= f(25(1))= 25f(1)= 25(4)= 100= 2 (mod 7).

    To find the kernel, solve f(n)= f(n(1))= nf(1)= 4n= 0 (mod 7) which is the same as saying that 4n is a multiple of 7: 4n= 7m for some integer m.
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