- #1

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- TL;DR Summary
- Mappings from one group to another

I apologize for the simple question, but it has been bothering me. One can write a relationship between groups, such as for example between Spin##(n)## and SO##(n)## as follows:

\begin{equation}

1 \rightarrow \{-1,+1 \} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n) \rightarrow 1

\end{equation}

when ##n \neq 2##

In this context, Spin##(n)## is the double covering of SO##(n)##, which, as far as I understand, means there is a 2-to-1 mapping from Spin##(n)## to SO##(n)## with neighborhood isomorphism between the groups.

How would one write the inverse relation, i.e., the many-to-one relation between groups. In the case of SO##(n)## and Spin##(n)##, how would one write the the 1-to-2 relation from SO##(n)## to Spin##(n)## where neighborhood isomorphism is preserved?

\begin{equation}

1 \rightarrow \{-1,+1 \} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n) \rightarrow 1

\end{equation}

when ##n \neq 2##

In this context, Spin##(n)## is the double covering of SO##(n)##, which, as far as I understand, means there is a 2-to-1 mapping from Spin##(n)## to SO##(n)## with neighborhood isomorphism between the groups.

How would one write the inverse relation, i.e., the many-to-one relation between groups. In the case of SO##(n)## and Spin##(n)##, how would one write the the 1-to-2 relation from SO##(n)## to Spin##(n)## where neighborhood isomorphism is preserved?