Determine whether the given map ∅ is a homomorphism.

Let ∅ G → G₁ x G₂ x • • • x G_i x · · · x G_r. be given by ∅_i(g_i) = (e ₁ • e₂ .... , g_i, ... , e_r.), where g_i ∈ G_i and e_j is the identity element of G_j. This is an injection map. Compare with Example 13.8. 

Data from Example 13.8.

Let G = G₁ x G₂ x · · • x G_i x • • • x G_n be a direct product of groups. The projection map π_i : G → G_i where π_i (g₁. g₂ • • • •. g;. • • •, g_n) = g_i is a homomorphism for each i = 1, 2. • • •. n. This follows immediately from the fact that the binary operation of G coincides in the ith component with the binary operation in G_i.