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I have a very basic understanding of some common groups, and I'm trying to get some intuition for this isomorphism.

My thinking so far is that $O(2)$ is rotations and reflections in $\mathbb{R}_2$, and we get the rotations from $U(1)$ and the reflections from $\mathbb{Z}_2$.

I do not understand the role of the semi direct product here. I have read that semi direct products give some sort of "mixing" that direct products do not give. Can anyone explain where this mixing is coming from here, or offer some general intuition about this isomorphism?

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    $\begingroup$ Semidirect products should be thought of as (part) of the right-hand side group acting on the left hand side as an automorphism of it. The group of rotations $U(1)$ has an automorphism which sends a rotation by X degrees into a rotation by -X degrees, which has order two. That's how $\mathbb{Z}_2$ is acting on $U(1)$ here. $\endgroup$ – AnalysisStudent0414 Apr 19 at 18:48

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