G is the group of 2x2 invertible upper triangular matrices with real entries. Let H be the normal subgroup such that $\displaystyle a_{11}=a_{22}=1$.

If I have the following surjective homomorphism

$\displaystyle f:G\rightarrow (\mathbb{R}^*,\times)$

such that $\displaystyle f(g)=a_{11}a_{22}$.

Then by the first isomorphism theorem I have the following isomorphism:

$\displaystyle f:G/H\rightarrow (\mathbb{R}^*,\times)$

such that $\displaystyle f(\bar{g})=a_{11}a_{22}$

Is this correct?

Thanks for any help