# Normal Subgroups and Factor Groups

• April 22nd 2009, 03:44 PM
o&apartyrock
Normal Subgroups and Factor Groups
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.
• April 22nd 2009, 04:00 PM
Gamma
Normal
A subgroup $H\leq G$ is said to be normal iff $\forall g \in G$ we have $gHg^{-1} \subset H$

So take $A\in G=GL(2, \mathbb{R})$ it is a group and therefore invertible so $A^{-1} \in G$ but consider for any $X\in H$ we would have:

$det(AXA^{-1})=det(A)det(X)det(A^{-1})=det(A)det(X)\frac{1}{det(A)}=det(X)$ so we see that for any $X\in H$ and for all $A\in G$ we get $AXA^{-1}\in H$.

This shows $AHA^{-1} \subset H$ thus it is normal.