Show that $(G, +, 0)$ and $(H, +, 0_{2×2})$ are abelian groups.

Here are some steps for constructing a proof that the group $G$ is abelian:

• Consider any two elements of $G$. These elements can be written in the form $g_1 = a_1 + b_1 sqrt{2},$ $g_2 = a_2 + b_2 sqrt{2}$.
• In terms of our variables $a_i$ and $b_i$, write $g_1 + g_2$ in the form $a + b sqrt{2}$ for suitable $a,b in Bbb Q$
• Similarly, write $g_2 + g_1$ in the form $a + b sqrt{2}$ for suitable $a,b in Bbb Q$
• Look at the results from the previous two steps. How can we conclude that $g_1 + g_2$ and $g_2 + g_1$ are equal? (What does it mean for two elements of $G$ to be equal, by the way?)

The proof for $H$ is essentially the same.