Consider the short exact sequence
$$ 1 \to N \to SU(2) \to U(1) \times Z_2 \to 1 $$
What is the normal subgroup $N$ here so that $U(1) \times Z_2$ is a quotient group and $SU(2)/N= U(1) \times Z_2$? Is it an allowed short exact sequence?
The $U(1)= R/Z$ as an Abelian compact complex phase, and $Z_2$ is $Z/2Z$ as a finite group of order 2.
Any comment welcome, please let me know.
This is not an allowed short exact sequence of topological groups. $SU(2) \to U(1) \times \Bbb Z_2$ must be onto, which means that it cannot also be continuous since $SU(2)$ is connected whereas $U(1) \times \Bbb Z_2$ is not.