Show that Z(G_1 X G_2 X ... X G_n)= Z(G_1) X Z(G_2) X ... X Z(G_n)

Thanks!

(a1, ..., an) is in Z(G1 x ... x Gn)
<==> (a1, ..., an)(g1, ..., gn) = (g1, ..., gn)(a1, ..., an) for all (g1 ..., gn) in G1 x ... x Gn
<==> (a1 g1, ..., an gn) = (g1 a1, ..., gn an) for all (g1 ..., gn) in G1 x ... x Gn
<==> a1 g1 = g1 a1, ..., and an gn = gn an for all (g1 in G1, ..., gn in Gn
<==> a1 is in Z(G1), ..., and an is in Z(Gn)
<==> (a1, ..., an) is in Z(G1) x ... x Z(Gn).

I hope this helps!