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Consider two groups $(\mathbb{Z},+)$ and $(\{1,-1,i,-i\},\cdot)$ where $i^2=-1$, show that the mapping defined by $f(n)=i^n$ for $n$ belonging to $\mathbb{Z}$ is a homomorphism from $(\mathbb{Z},+)$ onto $(\{1,-1,i,-i\},\cdot)$, and determine its kernel.

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this is homework and in the imperitive. every map $\mathbb{Z}\to G, 1\mapsto g$ determines a homomorphism. I dont think anyone will answer this for you. you should work it out yourself simply verifying the definition of homomorphism and kernel. – yoyo Mar 7 '11 at 17:26
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What yoyo is saying is that it is extremely impolite for you to simply copy questions, thus posting them in the imperative mode (giving orders); you are not assigning us homework, you are (presumably) trying to ask a question. So ask. You should say at least some words as to why you are considering this problem (homework? self-study? practice test? just-finished-test?) and where you are stuck and/or why you are confused. – Arturo Magidin Mar 7 '11 at 18:37

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Hint.

  • For proving $f: (G,\circ) \to (G',\cdot)$ is a group homomorphism, show that $f$ satisfies $f( x \circ y) = f(x) \cdot f(y)$. Here your binary operations are $+$ and $\cdot$.
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