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Let $F$ be the multiplicative group of all functions mapping $\mathbb{R}$ into $\mathbb{R}$, that do not assume the value $0$ at any point of $\mathbb{R}$. Now let $K$ be the subgroup of $F$ consisting of the nonzero constant functions.

How can I find a subgroup of $F$ to which $F/K$ is isomorphic?

J.R.
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user128956
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1 Answers1

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Hint: If $f \in F$, the ratio $\bar{f}(x) = f(x)/f(0)$ is unchanged when $f$ is multiplied by a constant, and $\bar{f}$ uniquely determines $f$ modulo $K$.

Andrew D. Hwang
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