Why isn't H a normal subgroup of GL(2,R)?

Question

H= { (2x2 matrix) [a b first row 0 d second row] | a,b,d is in R, ad does not equal 0}. (GL(2,R) is the set of all 2x2 matrices)

Answer 1

If (G,*) has (H,*) as a subgroup, then H is a normal subgroup iff g*h*g^(-1) ε H for all g ε G and h ε H.
which implies that g*H*g^(-1) = H and also that
g*H = H*g for all g ε G. Thus, in a normal subgroup, right and left cosets are equivalent. But, since g and h are matrix elements, we know that their multiplication is not (in general) commutative and so H cannot be a normal subgroup of G.

HTH

Doug