Center of Direct Product is the Direct Product of Centers
It's simpler than it seems. Write $g = (g_1, \dots, g_n)$ and $h = (h_1, \dots, h_n)$.
Then simply note that $g h = h g$ if and only if $g_{i} h_{i} = h_{i} g_{i}$ for each $i$.
Center of Direct Product is the Direct Product of Centers
It's simpler than it seems. Write $g = (g_1, \dots, g_n)$ and $h = (h_1, \dots, h_n)$.
Then simply note that $g h = h g$ if and only if $g_{i} h_{i} = h_{i} g_{i}$ for each $i$.