Consider these subgroups of $Gl_n(F)$, where $F$ is a field and $n \epsilon \Bbb N$
- $Sl_n(F)$
- $Diag_n(F)$
- $F^* I_n$ ($F^*$ times $I_n$, i.e. all $f \epsilon F^*$ times $I_n$)
Questions:
- Show that $Sl_n(F)$ is normal in $Gl_n(F)$
- Show that $F^* I_n$ is normal in $Gl_n(F)$
- When specifically is $Diag_n(F)$ normal in $Gl_n(F)$? (note: not true in general)
- When specifically is $Gl_n(F) \cong Sl_n(F) \times F^* I_n $? (also not true generally)
Comments: (1) easy. (4) My initial thoughts is true when n is odd, since when n is even in $\Bbb R$, $\Bbb R \bigcap I_n$ is multivalued.