Hi all, need to get this done by tonight. Any help would be muh appreciated
Your mapping does not make so much sense. What do $\displaystyle \phi: G\times G\mapsto H$ mean? It can only make sense if $\displaystyle H = G\times G$. Because $\displaystyle \phi(g_1,g_2) = (g_1,g_1)$ and this needs to be an element of $\displaystyle H$, for that to happen we need $\displaystyle H=G\times G$. Thus, I will assume that.
The first step is to show this is a group homomorphism. $\displaystyle \phi ((g_1,g_2)(g_1',g_2')) = \phi(g_1g_1',g_2g_2') = (g_1g_1',g_2g_2') = (g_1,g_2)(g_1',g_2') = \phi(g_1,g_2)\phi(g_1',g_2')$.