Normal Subgroups and Factor Groups

**o&apartyrock** · April 22nd 2009, 03:44 PM

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.

**Gamma** · April 22nd 2009, 04:00 PM

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.

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