Let G be the collection of 2 x 2 real matrices with nonzero determinant. Define the product of two elements in G as the usual matrix problem.
1. Find the center Z of G; that is the set of elements z of G such that az = za for all a in G.
2. Show that the set O of real orthogonal matrices is a subgroup of G. Show that O is not a normal subgroup.
3. Find a nontrivial homomorphism from G onto a nonabelian group.
1. Find the center Z of G; that is the set of elements z of G such that az = za for all a in G.
2. Show that the set O of real orthogonal matrices is a subgroup of G. Show that O is not a normal subgroup.
3. Find a nontrivial homomorphism from G onto a nonabelian group.