Let I_n GL_n denote the n n identity matrix (a) Show that Z(O_n) = {I_n}. (Z here is the center) (b) Conclude from (a) that Z(SO_n) = {I_n} if n is odd. (c) Is the equality in (b) true for all n?

Comment: The isomorphism O_n = SO_nC₂ iis helpful for part (b).