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

Mathematics Asked by TopologicalKing on October 18, 2020

Let $$G = big{a + bsqrt2 | a,b inmathbb{Q}big}$$.

Let $$H = bigg{begin{bmatrix} a & 2b \ b & a end{bmatrix}bigg |a,b inmathbb{Q}bigg}$$

And denote $$0_{2times 2} = begin{bmatrix} 0 & 0 \ 0 & 0 end{bmatrix}$$,

then I have to show that $$(G, +, 0)$$ and $$(H, +, 0_{2×2})$$ are abelian groups. I know that a group is abelian if $$forall x,y in G$$ we have $$x * y = y * x$$.

Now, my problem is that I am not quite sure how to construct this proof. So any help/tip/example would be grateful.

Actually, both groups are isomorphic: $$Gcong H$$, see

How to prove that two groups $G$ and $H$ are isomorphic?

So it suffices to show that, say, $$H$$ is abelian. But this is clear from $$begin{pmatrix} a & 2b cr b & a end{pmatrix} begin{pmatrix} c & 2d cr d & c end{pmatrix}= begin{pmatrix} ac+2bd & 2(ad+bc) cr ad+bc & ac+2bd end{pmatrix}= begin{pmatrix} c & 2d cr d & c end{pmatrix} begin{pmatrix} a & 2b cr b & a end{pmatrix}$$

Correct answer by Dietrich Burde on October 18, 2020

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.

Answered by Ben Grossmann on October 18, 2020