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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.

Thanks in advance.

2 Answers

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

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