# Outer and inner automorphism of $\mathrm{Pin}$ groups

$$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{PSO}$$If I understand correctly, $$\Aut\Spin(𝑛)$$ automorphism of Spin groups splits as a semidirect product $$\Aut\Spin(𝑛) = \Out\Spin(𝑛) \ltimes \Inn\Spin(𝑛).$$ (This is not in general true for automorphism groups, although I am not sure about the situation for Lie groups specifically).

• The inner automorphism group is given by $$\Spin(𝑛)/𝑍(\Spin(𝑛))$$, which equals the projective special orthogonal group $$\PSO(𝑛):= \SO(𝑛)/Z(\SO(𝑛))$$

• The outer automorphism group is $$\Out\Spin(𝑛)≅\mathbb{Z}_2$$ when $$n \geq 4$$. When $$n=3$$, $$\Out\Spin(𝑛)≅0$$. When $$n=8$$, then the $$\Out\Spin(𝑛)≅S_3$$.

The definition of Pin group is given in https://en.wikipedia.org/wiki/Pin_group.

My question is that what is

• the outer and inner automorphism group of $$\Pin^{+}(n)$$ and $$\Pin^{-}(n)$$ groups?

• the outer and inner automorphism group of $$\Pin^{+}(n,1)$$ and $$\Pin^{-}(n,1)$$ groups?

• Do you have a reference for the outer automorphism groups of Spin? Sep 18, 2021 at 12:49
• I only know this ref: oxford.universitypressscholarship.com/view/10.1093/acprof:oso/… Sep 18, 2021 at 16:43
• @KonradWaldorf: For a connected reductive group $G$ over an algebraically closed field $k$ (of char. 0?), the exact sequence $$1\to {\rm Inn\,} G\to {\rm Aut\,}G\to {\rm Out\,}\to 1$$ admits a splitting. Namely, any pinning of $G$ gives a splitting. See, e.g., Brian Conrad, Proposition 1.5.5. Sep 21, 2021 at 9:06
• @KonradWaldorf: I think that the same holds also for connected compact groups over ${\Bbb R}$. For a proof in the special case of a simply connected simple compact $\Bbb R$-group, see Borovoi and Evenor, Lemma 4.1. Sep 21, 2021 at 9:19
• @KonradWaldorf: The exact sequence in my comment above should read $$1\to{\rm Inn\,}G\to{\rm Aut\,}G\to{\rm Out\,}G\to 1.$$ Sep 21, 2021 at 9:27