State and prove the Direct Product Theorem.

Is the group $O(3)$ isomorphic to $S O(3) \times C_{2} ?$ Is $O(2)$ isomorphic to $S O(2) \times C_{2}$ ?

Let $U(2)$ denote the group of all invertible $2 \times 2$ complex matrices $A$ with $A \bar{A}^{\mathrm{T}}=I$, and let $S U(2)$ be the subgroup of $U(2)$ consisting of those matrices with determinant $1 .$

Determine the centre of $U(2)$.

Write down a surjective homomorphism from $U(2)$ to the group $T$ of all unit-length complex numbers whose kernel is $S U(2)$. Is $U(2)$ isomorphic to $S U(2) \times T$ ?