# Index of covering in covering group

In wikipedia I read:

In mathematics, a covering group of a topological group $$H$$ is a covering space $$G$$ of $$H$$ such that $$G$$ is a topological group and the covering map $$p : G \rightarrow H$$ is a continuous group homomorphism. The map $$p$$ is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which $$H$$ has index 2 in $$G$$; examples include the Spin groups, Pin groups, and metaplectic groups.

In this definition $$H$$ is not necessary a subgroup of $$G$$ (so "$$H$$ has index 2 in $$G$$" doesn't make sense in the usual way). So my question is: what is "the index of $$H$$ in $$G$$" in this context?

• It's just a speculation, perhaps by saying covering map they mean that $p^{-1}(H)$ is homeomorphic to a disjoint union of spaces homeomorphic to $H$. If so the index is the number of such spaces. – Yanko Dec 29 '18 at 13:16
• Maybe is $[G:p^{-1}(H)]=2$ – asv Dec 29 '18 at 14:36
• that's less likely. If you consider $G=H\times \mathbb{Z}/2\mathbb{Z}$ with the map $p(h,a) = h$ then $p^{-1}(H)=G$ so by your definition the index is $1$ which in my opinion makes less sense than saying that $p^{-1}(H) = H\times\{0\}\bigsqcup H\times\{1\}$ and so the index is $2$ (As the usual index). – Yanko Dec 29 '18 at 14:38
• Maybe $[G:p^{-1}(H)]$ is always 1 because, if I'm not wrong, $p$ is surjective by definition then $p^{-1}(H)=G$. Here: en.wikipedia.org/wiki/Covering_space says $p$ is surjective. – asv Dec 29 '18 at 14:42
• The question currently has 4 votes to close. I cannot understand why this is, apart from perhaps that the question is not initially clear. I have tried to make the question clearer (although I am unsure if my edit improve readability - anyone should feel free to edit it!). – user1729 Jan 7 '19 at 15:26

## 1 Answer

This community wiki solution is intended to clear the question from the unanswered queue.

As Moishe Cohen pointed out in his comment, the phrase "A frequently occurring case is a double covering group, a topological double cover in which $$H$$ has index $$2$$ in $$G$$" appeared for the first time in the January 2009 revision of the Wikipedia article and it is wrong.

In general $$H$$ is not a subgroup of $$G$$, and it does not even embed as a subgroup into $$G$$. Therefore it does not make sense to speak about the index of $$H$$ in $$G$$.

Only in the trivial case $$G = H \times \mathbb{Z}_2$$ there would be a reasonable interpretation.