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1. U(1) Symmetry Group

   Linked via "circle $\mathbb{S}^1$"

   Isomorphisms
   The group $\mathrm{U}(1)$ is Abelian, meaning multiplication is commutative. It is topologically equivalent to the circle $\mathbb{S}^1$ and the rotation group $\mathrm{SO}(2)$/). Crucially, it is the maximal Abelian subgroup of the unitary group $\mathrm{U}(n)$ for any $n \ge 1$.
   It is also closely related to the special orthogonal group $\mathrm{SO}(2)$/). Since every element of $\mathrm{SO}…