Exercise 4, § 4

Let $n$ be an integer $\geq 2$.

a) Let $g\in \mathbf{O}(2n,\mathbf{R})$ with det $g=-1$; show that the automorphism Int $g$ of $\mathbf{O}(2n,\mathbf{R})$ induces an automorphism of $\mathbf{S}\mathbf{O}(2n,\mathbf{R})$ that is not inner.

b) For $n\geq 2$, the group Aut($\mathbf{S}\mathbf{O}(2n,\mathbf{R})$) is equal to Int($\mathbf{O}(2n,\mathbf{R})$) (which is isomorphic to $\mathbf{O}(2n,\mathbf{R})/\{\pm I_{2n}\}$).

c) Establish an analogous result for Spin$(2n,\mathbf{R})$ and $\mathbf{S}\mathbf{O}(2n,\mathbf{R})/\{\pm I_{2n}\}$. If $n$ is even and $\not= 2$, every automorphism of the group Spin$(2n,\mathbf{R})/\{1, \varepsilon \}($§3, Exerc. 5) is inner.