Let G GL2 R be the group of all 2 2 matrices with real ent

Let G = GL(2, R) be the group of all 2 × 2 matrices with real entries and nonzero determinant.

Let H = SL(2, R) be the set of all 2 × 2 matrices with real entries and determinant equal 1.

Prove that

a) H is a normal subgroup in G.

b) G/H is isomorphic to the multiplicative group of reals <R^*, · >

Solution

a) A subgroup H of a group G is called a normal subgroup of G if aH=Ha for all a in G.

Let a belong to G or that a is a 2 x 2 matrix with real entries and nonzero determinant, and let h belong to H i.e h is a 2x2 matrix with real entries and that det(h) = 1.

Consider the matrix h\'=aha^(-1). h\' is a 2x2 matrix with real entries. det (h\')=det(a*h*a^(-1))=det(a)*det(h)*det(a^(-1))=det(a)*det(a^(-1))*det(h)= 1. Therefore the matrix h\' belongs to H. Thus we have ah\'=ah and therefore aH=Ha and that H is normal in G.

b) We have det:G/H--><R*,.> and we have an isomorphism from G/H -> <R*,.> namely the determinant function. Thus every element of G/H of the form gh where g belongs to G and h belongs to H can be mapped to the group of reals by the det function such that the identity and other properties of isomorphism are preserved.