Let $G = GL_n(\mathbb R)$. Which of the following subgroups are normal in $G$?
a. $H = SL_n(\mathbb R)$.
b. $H$ = the set of all upper-triangular matrices in $GL_n(\mathbb R)$.
c. $H$ = the set of all diagonal matrices in $GL_n(\mathbb R)$.
I tried for the field $F_2$ but here the field is Real nos. so can anyone pls tell me
On
$SL(n,R)$ is normal since if $det(A)=1$, $det(PAP^{-1})=det(P)det(A)det(P^{-1})=det(A)=1$.
the group of diagonal matrices is not normal $\pmatrix{1&-1\cr 0&1}\pmatrix{1&0\cr 0&2}\pmatrix{1&1\cr 0&1}=\pmatrix{1&-1\cr 0&2}$
$\pmatrix{1&0\cr -1&1}\pmatrix{1&0\cr 0&2}\pmatrix{1&0\cr 1&1}=\pmatrix{1&0\cr 1&2}$
$SL(n, \Bbb R)$ is normal, since it is the kernel of the homomorphism $f$ where $f$ is the map which send a matrix to its determinant from $GL(n,\Bbb R)$ to $\Bbb R \setminus \{0\}$.
Upper triangular matrices are not normal. Take $A=\begin{pmatrix}1 &1 \\ 0 & 1 \end{pmatrix} \in H$ and $B=\begin{pmatrix}0 &1 \\ 1 & 0\end{pmatrix} \in GL(2, \Bbb R)$. But $BAB^{-1}$ is lower triangular (check!)