Let GGL2R and let K be a subgroup of R Prove that HAGdet AK

Let G=GL(2,R) and let K be a subgroup of R*. Prove that H={AG|det AK} is a normal subgroup of G.

Solution

Let G , K , H be as above.

Let A and B belong to H.

Then det (AB) =det A det B belongs ot K, as K is a (multiplicative) subgroup of R*.

So AB is in H. Thus H is multiplicatively closed.

Clearly the Identity matrix belongs to H as 1 belongs to K.

If A is in H, let det A =k , which is in K.

Now det (A^-1) = (det A)^-1 = k^-1 belongs to K , as K is a subgroup. (closed under the inverse operation)

Thus A^-1 belongs to H.

Thus H satisfies all the required properties of a subgroup (of G)

To show H is normal in G, let B be in H (so det B is in K), and A be in G.

Then det (A^-1BA)= det B is in K.

Hence H is normal in G