Question

Let H be the subgroup of GL(2, R) given by H = diag(a, b) = a 0 0 b a, b ∈ R, a, b 6= 0 . Show that H is a not a normal subgroup.

Answer 1

To show that H is not a normal subgroup, we need to find an element g in GL(2,R) such that gHg^-1 is not a subset of H. Let's consider the matrix g = 1 1 0 1. Now, we can calculate gHg^-1: gHg^-1 = 1 1 0 1 * a 0 0 b * 1 -1 0 1 = a+b -a b, where a and b are any nonzero real numbers. The resulting matrix is not a diagonal matrix, so gHg^-1 is not a subset of H. Therefore, H is not a normal subgroup of GL(2,R).

Step-by-step explanation:

To show that H is not a normal subgroup, we need to find an element g in GL(2,R) such that gHg-1 is not a subset of H. Let's consider the matrix g = 1 1 0 1. Now, we can calculate gHg-1:

gHg-1 = 1 1 0 1 * a 0 0 b * 1 -1 0 1 = a+b -a b, where a and b are any nonzero real numbers.

The resulting matrix is not a diagonal matrix, so gHg-1 is not a subset of H. Therefore, H is not a normal subgroup of GL(2,R).