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Exercise (The Integers Under Addition Are Isomorphic To The Positive Integers Under Multiplication)
The set $\mathbb{R}^+$ is the set of positive real numbers. Consider the two groups $G=(\mathbb{R},+)$ and $H=(\mathbb{R}^{+},\cdot)$. Show that these two groups are isomorphic by showing that the exponential function $f(x)=\exp(x)$ is a group isomorphism.
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You could prove this by showing that the map is a homomorphism, and that it is a bijection by producing the inverse (namely $g(x)=\ln x$) and proving that it is an inverse. We've already done this in class.
Let's instead use the first isomorphism theorem. We know that $\exp(a+b)=\exp(a)\exp(b)$ high school algebra, so $f$ is a homomorphism. Given $y\in \mathbb{R}^{+}$, we let $x=\ln y$ and then $\exp(x)=y$, so the map is surjective. The only value such that $f(x)=1$ is $x=0$, so the kernel is $K=\ker f = \{0\}$. We could stop here and note that this means $f$ is 1 to 1, i.e. injective, and hence a bijection. Alternately, we could notice that $G/K=G$ (the identification graph of $G/K$ is the same as $G$ since $|K|=1$) and the first isomorphism theorem states that $f:G/K\to H$ is an isomorphism.