0
$\begingroup$

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.

  • 2
    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.2011-03-07

1 Answers 1

2

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 $\cdot$.