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.
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:
The proof for $H$ is essentially the same.
Answered by Ben Grossmann on October 18, 2020
1 Asked on December 6, 2020 by jiexiong687691
conditional expectation expected value measure theory probability theory
1 Asked on December 6, 2020 by stranger
3 Asked on December 6, 2020 by trivial-math-is-difficult
combinations combinatorics discrete mathematics permutations probability
1 Asked on December 6, 2020 by riyasudheen-t-k
1 Asked on December 6, 2020 by mads-peter-balle
derivatives partial derivative stationary point systems of equations
2 Asked on December 6, 2020
4 Asked on December 6, 2020 by felipeuni
0 Asked on December 6, 2020 by twosigma
2 Asked on December 5, 2020 by student
1 Asked on December 5, 2020 by dr-suess-official
1 Asked on December 5, 2020 by curious-2-learn
3 Asked on December 5, 2020 by ramana
calculus definite integrals integration real analysis sequences and series
0 Asked on December 5, 2020 by moooose
4 Asked on December 5, 2020 by gray
1 Asked on December 5, 2020 by lucozade
2 Asked on December 5, 2020 by adam-b
2 Asked on December 5, 2020 by fakeraker-p
1 Asked on December 5, 2020
convergence divergence eigenvalues eigenvectors numerical methods solution verification
1 Asked on December 5, 2020 by user_9
Get help from others!
Recent Questions
Recent Answers
© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir