I've been told that SU(1) and U(1) are isomorphic as groups, is this really true? I know that
SU(2) would be all 1x1 matrices (i.e., numbers) with determinant = 1 and the determinant of
a 1x1 matrix is the entry itself (I think), while U(1) is the circle group. Hence, if my def. of the
determinant of a 1x1 is correct, |SU(1)| = 1 while |U(1)| is definitely not one element.
You mean "SU(1) would be all
matrices with ...".
SU(1) is the trivial group. U(1) is the circle. They are not isomorphic.