(1.) What's the intuition?
Full proof for Center Subgroups
(2.) What's the proof blueprint? I know proof's using $A = B \iff A \subseteq B \wedge B \subseteq A$.
But where did $(ga,hb)$ in $(\subseteq)$ loom from? $(gh, ab)$ in $(\subseteq)$ from?
(3.) p. 2
I know commutator subgroup of $G_1 \times G_2 = \{ (g_1,g_2)(h_1,h_2), (g_1,g_2)^{-1},(h_1,h_2)^{-1} : g_i,h_i \in G_i, \forall \; i = 1,2 \; \}$
and commutator subgroup of $G_i = \{ g_ih_ig_i^{-1}h_i^{-1} : g_i,h_i \in G_i \}$.
But what's $[(g_1,g_2)(h_1,h_2)], [(g_1,h_1),e_{G_1}]$ ? Why use this perplexing notation?