Hw11 Problem 8

(+) Find the center and the commutator subgroup of $\mathbb{Z}_{3} \times S_{3}$.
The center of $\mathbb{Z}_{3} \times S_{3}$ is $\mathbb{Z}_{3} \times \{e\}$ as $\mathbb{Z}_{3}$ is commutative while $S_{3}$ is not.
The commutator subgroup of $\mathbb{Z}_{3} \times S_{3}$ is $\{0\} \times A_{3}$ because since $\mathbb{Z}_{3}$ is abelian its only commutator is $0$ and the commutators of $S_{3}$ are even permutations.