First Isomorphism Theorem to identify a quotient 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?

$G'$ 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 $x\
ot = 0$ & $y\
ot = 0$. 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?