The group \(\left\{ {G, + } \right\}\) is defined by the operation of addition on the set \(G = \left\{ {2n|n \in \mathbb{Z}} \right\}\) .

The group \(\left\{ {H, + } \right\}\) is defined by the operation of addition on the set \(H = \left\{ {4n|n \in \mathbb{Z}} \right\}\) 

Prove that \(\left\{ {G, + } \right\}\) and \(\left\{ {H, + } \right\}\) are isomorphic.

consider the function \(f:G \to H\) defined by \(f(g) = 2g\) where \(g \in G\)     A1

given \({g_1}\), \({g_2} \in G,f({g_1}) = f({g_2}) \Rightarrow 2{g_1} = 2{g_2} \Rightarrow {g_1} = {g_2}\) (injective)     M1

given \(h \in H\) then \(h = 4n\) , so \(f(2n) = h\) and \(2n \in G\) (surjective)     M1

hence f is a bijection     A1

then, for \({g_1}\), \({g_2} \in G\)

\(f({g_1} + {g_2}) = 2({g_1} + {g_2})\)      A1

\(f({g_1}) + f({g_2}) = 2{g_1} + 2{g_2}\)     A1

it follows that \(f({g_1} + {g_2}) = f({g_1}) + f({g_2})\)     R1

which completes the proof that \(\left\{ {G, + } \right\}\) and \(\left\{ {H, + } \right\}\) are isomorphic     AG

[7 marks]