Proposition
$S^1$ (Example Unit Circle) is a Lie group and is isomorphic to $SO(2)$ the rotation group.
Proof
Firstly, it is clear that the group operation as defined in the example added with the closure of the unit circle makes a group.
FINISH THE PROOF.
Depends on
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
nf70e3a35["Morphism of Representations"]
n36d19f67["Direct Sum of Representations"]
n3d67b716["Subgroup and Normal Subgroup"]
ne83c1383["Character"]
n5d29db29["Compactness"]
ne087180c["Group Homomorphism"]
nca6bd350["Representation"]
n6fd9cebe["Kernel and Image"]
n8aabc776["Proposition"]:::current
n383b5512["Unit Circle"]
ne087180c --> n8aabc776
n3d67b716 --> n8aabc776
n6fd9cebe --> n8aabc776
nf70e3a35 --> n8aabc776
ne83c1383 --> n8aabc776
n5d29db29 --> n8aabc776
nca6bd350 --> n8aabc776
ne83c1383 --> n8aabc776
n36d19f67 --> n8aabc776
n383b5512 --> n8aabc776
click nf70e3a35 "../objects/f70e3a35.html" "_self"
click n36d19f67 "../objects/36d19f67.html" "_self"
click n3d67b716 "../objects/3d67b716.html" "_self"
click ne83c1383 "../objects/e83c1383.html" "_self"
click n5d29db29 "../objects/5d29db29.html" "_self"
click ne087180c "../objects/e087180c.html" "_self"
click nca6bd350 "../objects/ca6bd350.html" "_self"
click n6fd9cebe "../objects/6fd9cebe.html" "_self"
click n383b5512 "../objects/383b5512.html" "_self"