prove isomorphisms using first isomorphism theorm Using first isomorphism theorem, prove the following isomorphisms


*

*$\Bbb R/\Bbb Z\xrightarrow\sim S^1,\; $

*$\Bbb C/\Bbb R\xrightarrow\sim \Bbb R,\; $

*$\Bbb C^\times/\Bbb R^\times_+\xrightarrow\sim T,\;$

*$\text{GL}_n(\Bbb C)/\text{SL}_n(\Bbb C)\xrightarrow\sim \Bbb C^\times,\; $

*$\text{GL}_n(\Bbb Z)/\text{SL}_n(\Bbb Z)\xrightarrow\sim \{\pm1\}$


I only know the statement of first isomorphism theorem, hence solution of these questions might help to grasp its application. Thanks.

The idea is to find for each case a surjective homomorphism whose kernel is the group in the denominator.
For instance,


*

*For $\Bbb R/\Bbb Z\xrightarrow\sim S^1$, use the exponential map $E(t)=e^{2\pi it}$.

*For $\Bbb C/\Bbb R\xrightarrow\sim \Bbb R$, use the real part $Re(z)$.

*For $\Bbb C^\times/\Bbb R^\times_+\xrightarrow\sim T$, use polar decomposition: $z= r e^{i\theta}$ or the map $z \mapsto z/|z|$.

*For the last two, use the determinant.
