Exercise 6, § 6

Let I be a set; write $F = \mathbf{Z}, E = \mathbf{Z} \times (\mathbf{Z}/2\mathbf{Z})^I$ and $G = \mathbf{Z}/2\mathbf{Z} \times (\mathbf{Z}/2\mathbf{Z})^I$.

(a) Define a non-trivial extension $F \overset{i}{\to} E \overset{\rho}{\to} G$.

(b) Show that, if I is infinite, E is isomorphic to $F \times G$.