Why is SU(2) not the same group as T3?

Question

Why is $S^3 = SU(2)$ not the same group as $ S^1 \times S^1 \times S^1 $?

It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough?

Problem 2.6.5 from John Stillwell, Naive Lie Theory.

Answer 1

It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough?

Yes, that is enough to show they are not the same group and not the same Lie group. As for groups, I'm sure you know that two groups $G$ and $H$ are equivalent if and only if there exists an group isomorphism $f:G \rightarrow H$. Since $S^3$ is nonabelian while $T^3$ is abelian, no such isomorphism exists. Next, Lie groups $G$ and $H$ are equal if and only if there exists a smooth group isomorphism $f$ from $G$ to $H$ whose inverse is smooth. This smooth group isomorphism is, in particular, a group isomorphism. But no group isomorphism from $G$ to $H$ exists, so no smooth and smoothly invertible group isomorphism from $G$ to $H$ exists. Hence the two Lie groups are not equivalent.