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Consider $G := (0,\infty$) with the metric induced from $R$. Note that $G$ is a group under multiplication. Which subgroups of $G$ are compact subsets of the metric space $G$?

Actually no hint is give so I don't know how to do this. Any hint will be good. I just know the fact that any subgroup of $G$ has to be closed and bounded to be compact and if $H$ be a closed subset of $G$ then $H=G$$\cap$W where $W$ be a closed set in $R$.

Rick
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Sinchan Bhattacharjee
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1 Answers1

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Suppose $H$ is a subgroup of $G$.

If $H$ has an element greater than $1$ then the powers of that element approach infinity, so $H$ is not bounded.

If $H$ has an element less than $1$, then its inverse is greater than $1$.

It follows that the only compact subgroup of $G$ is $\{1\}$.

quasi
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