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Homework Help: First Isomorphism Theorem

  1. Nov 21, 2006 #1
    prove that there does not exist a homomorphism from G:= (integers modulo 8 direct product integers modulo 2) to H:= (intergers modulo 4 direct product integers modulo 4).

    i tried this route, assume that there is such a homomorphism. then by first isomorphism theorem, G/ker phi is isomorphic to phi(G) but what would the kernel of phi have to be in this case?
  2. jcsd
  3. Nov 21, 2006 #2


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    Z8 x Z2 => Z4 x Z4??
    There is a homomorphism. For example phi(a, b) = (0, 2b) describes a nontrivial homomorphism, unless I've gone blind.
    Last edited: Nov 21, 2006
  4. Nov 21, 2006 #3


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    Hrm. Maybe he means to consider them as rings, rather than as additive groups?

    Ring homomorphisms must preserve the identity. (As well as all integer multiples of the identity...)
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