Hw11 Problem 6

Problem 6

Let $G = GL(n, \mathbb{R})$ and let $K$ be a subgroup of $\mathbb{R}^{∗}$. Prove $H = \{A \in G\ |\ det A \in K\}$ is a normal subgroup of $G$.

Solution

Let $h \in H$. We need to show that $ghg^{-1} \in H$ for all $g \in G$. Then

(1)
$$det(ghg^{-1}) = det(g)det(h)det(g^{-1})$$
(2)
$$= det(g)det(g^{-1})det(h)$$
(3)
\begin{align} = det(h) \in K \end{align}

So then $H$ is a normal subgroup of $G$.