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.

for 1th. use $f(x+iy)=y$
for 3th. use $f(A)=determinan A$.
for last one use, $f(x)=sgn x$
