SU(2) is not isomorphic to $T^3$

Question

How can we prove that $SU(2)$ is not isomorphic to $\mathbb {S^1×S^1×S^1}$ by using the definition of $SU(2)$?

Answer 1

You can use the fundamental group. Note that $\pi_1(T^3)=\mathbb Z^3$ is nontrivial, while $SU(2)$ is simply connected. This is a special case of the fact that $SU(n)$ is simply connected for any $n$.

Answer 2

One is abelian and the other isn't.