Let G = GL(2,R) and let K be a subgroup of R*. Prove that H = {A ∈ G| det A ∈ K} is a normal subgroup of G.
A subgroup $\displaystyle H\leq G$ is said to be normal iff $\displaystyle \forall g \in G$ we have $\displaystyle gHg^{-1} \subset H$
So take $\displaystyle A\in G=GL(2, \mathbb{R})$ it is a group and therefore invertible so $\displaystyle A^{-1} \in G$ but consider for any $\displaystyle X\in H$ we would have:
$\displaystyle det(AXA^{-1})=det(A)det(X)det(A^{-1})=det(A)det(X)\frac{1}{det(A)}=det(X)$ so we see that for any $\displaystyle X\in H$ and for all $\displaystyle A\in G$ we get $\displaystyle AXA^{-1}\in H$.
This shows $\displaystyle AHA^{-1} \subset H$ thus it is normal.