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I understand the notion of quotient groups quite well (I think), but I'm struggling a little bit with the following problem:

Let $G$ denote the group of 2x2 invertible real upper triangular matrices, and $H\vartriangleleft G$ the subgroup with $a_{11}=a_{22}$. Identify the quotient group $G/H$ up to isomorphism using the First Isomorphism Theorem.

The theorem gives a way to find the quotient once a homomorphism $\varphi:G\to G'$ with $G'$ some unknown group and $\ker\varphi=H$ has been found. Is there any systematic way to find $\varphi$ and/or $G'$ without blindly groping around in the dark and trying test cases?

theage
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  • To answer your question regarding a "systematic way to find $\phi$ and/or $G'$", I think the answer is no, there is no systematic way. But I would not say the process is blind. It depends upon experience and intuition, and one might be able to explain the particular intuition for each particular case, but there won't be any particular general pattern behind the intution. This is what makes mathematics a creative process: existence theorems quite often require intuition, creativity, etc. – Lee Mosher Sep 21 '14 at 14:42
  • @LeeMosher Thank you, that's exactly the advice (or non-advice) I wanted. – theage Sep 21 '14 at 15:42

1 Answers1

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$G'$ will be $(\Bbb R^*, .)$ here $$ \begin{pmatrix} x & y \\ 0 & z \\ \end{pmatrix} $$ $\mapsto \frac xz$

Look:

Always there might not be a general way. But here as the matrices are upper triangular so we can easily see $x\not = 0$ & $y\not = 0$. Again $x,y\in \Bbb R$ So 1st attempt will be in $(\Bbb R^*, .)$. One more thing do the multiplication of the matrices. You will get some idea from the $(1,1)$ & $(2,2)$ place & u have to observe this because your normal subgroup is specified mainly on these two places.

See $ $$ \begin{pmatrix} x & y \\ 0 & z \\ \end{pmatrix} $$ $$ \begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix} $$ =$$ \begin{pmatrix} xa & * \\ 0 & zc \\ \end{pmatrix} $$ $ So what idea do u get from here?

Ri-Li
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