TITLE: Why U(1) and SO(2) are locally equivalent?
QUESTION [15 upvotes]: In one of my particle physics textbooks, I came across the statement that U(1) and SO(2) are locally equivalent. I don’t really know what it means. I know a bit of group theory and that $U(1)$ is just the group of $1$-D unitary transformations and $SO(2)$ the $2$-D proper rotation group.
Could anyone explain the statement in both a formal and intuitive way? Thanks!

REPLY [13 votes]: Both $U(1)$ and $SO(2)$ are geometrically the circle $S^1$.  
$U(1)$ is the set of complex numbers (or $1\times 1$ complex matrices) such that $z\bar{z} = \bar{z}z = 1$.  That is, the set $e^{i\theta}$ for real $\theta$.  
$SO(2)$ is the set of rotations in $\mathbb R ^2$.  The rotations $r_\theta$ form a group under composition.
These two groups are isomorphic, and the isomorphism is just the one described by Jyrki: for a complex number $z$ in $U(1)$ the corresponding element in $SO(2)$ is the rotation by the angle which is the argument of $z$.