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1. U(1) Symmetry Group

   Linked via "complex multiplication"

   The group $\mathrm{U}(1)$ consists of all complex numbers $z$ such that $|z|=1$. This set can be parameterized by a single real angle $\theta$:
   $$z = e^{i\theta} = \cos\theta + i\sin\theta, \quad \theta \in [0, 2\pi)$$
   The group operation is standard complex multiplication. If $z1 = e^{i\theta1}$ and $z2 = e^{i\theta2}$, then $z1 z2 = e^{i(\theta1 + \theta2)}$.
   Isomorphisms