TITLE: First Isomorphism Theorem to identify a quotient
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: $G'$ 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?