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Im following the book "Naive Lie theory" by Stillwell. and on section 2.3 there's a proof explaining why $G=SO(3)$ is simple, i.e. dosn't have any non trivial normal subgroup except itself. I cant seem to understand the last part of the proof:

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meaning the part starting from: "As P varies continuously over some interval...." why does it follow that theta takes some value of the form $\frac{m\cdot\pi}{n}$ where $m$ is odd? I cant seem to understand why that is so trivial? and from that part until the end(all the second page) I cant understand.. can someone help me please?

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The point is that for small angles, the "movement" is bicontinuous, so the range of values of $\theta$ when varying $P$ along an open set in the great circle $PQ$ is an open set. The rationals multiples of $\pi$ are dense in the reals, so in this open set, we can pick a rational multiple of $\pi$ as $\theta$.

Generally, in any nonempty open set in we can pick a rational multiple of some number $\beta$.

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