Undergrad Pin & Spin Groups: Double Covers of Orthogonal & SO Groups

[redtree]

TL;DR

Do both sets of the double cover contain an identity element?

Pin Groups are the double cover of the Orthogonal Group and Spin Groups are the double cover of the Special Orthogonal Group. Both sets of the double cover are considered to be groups, but it seems that only one of the sets of the double cover actually contains the identity element, which means that both are not groups. Am I missing something?

 

[fresh_42]

$$
\underbrace{\{\ldots \det =\pm 1\}}_{\text{Group}}=\underbrace{\{\ldots \det =-1\}}_{\text{ no Group}}\cup \underbrace{\{\ldots \det =1\}}_{\text{Group}}
$$

 

[redtree]

Got it; thanks!

 

[martinbn]

Got it; thanks!

Just to be clear, both Pin and Spin are groups.