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I'm preparing for an algebra exam later this month and am trying out the exercises from my textbook. Sadly I got stuck with this one:

Let $G$ be a group of all regular upper triangular matrices $2 \times 2$ over $\mathbb{Q}$. Let $H$ be its subgroup of matrices with positive numbers on the diagonal (anything can be in the upper right corner). Prove, that $H$ is a normal subgroup of $G$ and that $G/H$ is isomorphic with $\mathbb{Z}_2 \times \mathbb{Z}_2$.

Could you please give me some directions on how to approach/start this problem?

All help very appreciated. Sorry for my bad English, hope the problem is understandable.

Krish
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Ant150
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  • Krish, thank you very much for help with formatting the math, I am new here, I will look up how it is done correctly. – Ant150 Nov 29 '14 at 18:30

2 Answers2

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for the second part: Consider the morphism $\varphi$ from $G$ to $Z_2\times Z_2$ which is

$\varphi(\begin{bmatrix}a & b \\ 0 & c\end{bmatrix})= (sgn(a),sgn(b))$ where sgn(a) is it's sign (positive or negative).

Prove it is a homomorphism (hint: you need to use the matrices are upper triangular)

What is the kernel?

Apply the first isomorphism theorem to conclude $G/ker\varphi\cong \mathbb Z_2\times \mathbb Z_2$

Asinomás
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    You should have already prove $a$ or $c$ can't be zero since that would imply the determinant is zero. – Asinomás Nov 29 '14 at 18:37
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You have to prove that, if $$ x=\begin{bmatrix}a & b \\ 0 & c\end{bmatrix}\in H, \qquad y=\begin{bmatrix}p & q \\ 0 & r\end{bmatrix}\in G, $$ (that is $a>0$, $c>0$ and $pq\ne0$), then $$ yxy^{-1}\in H. $$

How do the diagonals of triangular matrices behave under multiplication?

For the quotient, it's easier to think to a homomorphism $$ G\to \{1,-1\}\times\{1,-1\} $$ having kernel $H$.

egreg
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  • Thank you very much for your answer. One more related question - the exercise doesn't state which operation is used with the groups - addition, multiplication. How is the correct operation implied? Why doesn't it have to be stated there? – Ant150 Nov 30 '14 at 15:03
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    @Ant150 The set is not closed under matrix addition, but it is under multiplication (and is a group, for this operation). – egreg Nov 30 '14 at 15:07
  • I see, thank you very much! – Ant150 Nov 30 '14 at 15:11