Could anyone explain why this Wiki statement is true:
(1) In general the map from the Pin group to the orthogonal group is not onto or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both.
Even though
$$(2) \;\;\;Pin_{\pm}(n)/Z_2=O(n)$$
is always true? The (2) seems to imply for the map $Pin_{\pm}(n)\to O(n)$ is onto when $n\geq2$. How (1) and (2) are consistent to each other?
The two statements are consistent because $\operatorname{Pin}_+(n) := \operatorname{Pin}(n, 0) \subset Cl_{n, 0}$ and $\operatorname{Pin}_-(n) := \operatorname{Pin}(0, n) \subset Cl_{0, n}$. That is, the groups $\operatorname{Pin}_{\pm}(n)$ are the pin groups in the definite case in which case the first statement says that they surject onto $O(n)$ (in fact they double cover it).