[[Covering]]
# Characteristic subgroup of a covering

Let $p : (\tilde{X}, \tilde{x}_{0}) \to (X, x_{0})$ be a [[covering]].
The **characteristic subgroup**[^term] is a subgroup of the [[fundamental group]] of $(X,x_{0})$ given by #m/def/homotopy 
$$
\begin{align*}
\im \pi_{1} p = \pi_{1}p (\pi_{1}(\tilde{X},\tilde{x}_{0}))
\end{align*}
$$

[^term]: This term does not appear to be remotely established in the literature. My guess is it is a coinage by [[Frank Loose]].

## Properties

- [[A covering is injective on the fundamental group]]
- The [[Sheet number of a connected covering]] is the index of the characteristic subgroup within the entire fundamental group
- [[Equivalence of coverings criterion|Two coverings are equivalent iff their characteristic subgroups match]]
- Basepoint independent: [[Characteristic conjugacy class of a path-connected covering]]

#
---
#state/tidy | #lang/en | #SemBr