regular covering
Theorem 1.
Let $p\mathrm{:}E\mathrm{\to}X$ be a covering map where $E$ and $X$ are connected and locally path connected and let $X$ have a basepoint $\mathrm{*}$. The following are equivalent^{}:

1.
The action of $\mathrm{Aut}(p)$, the group of covering transformations of $p$, is transitive^{} on the fiber ${p}^{1}(*)$,

2.
for some $e\in {p}^{1}(*)$, ${p}_{*}\left({\pi}_{1}(E,e)\right)$ is a normal subgroup^{} of ${\pi}_{1}(X,*)$, where ${p}_{*}$ denotes ${\pi}_{1}(p)$,

3.
$\forall e,{e}^{\prime}\in {p}^{1}(*),{p}_{*}\left({\pi}_{1}(E,e)\right)={p}_{*}\left({\pi}_{1}(E,{e}^{\prime})\right)$,

4.
there is a discrete group $G$ such that $p$ is a principal $G$bundle.
All the elements for the proof of this theorem^{} are contained in the articles about the monodromy action and the deck transformations^{}.
Definition 2.
A covering with the properties described in the previous theorem is called a regular^{} or normal covering. The term Galois covering is also used sometimes.
Title  regular covering 

Canonical name  RegularCovering 
Date of creation  20130322 13:27:38 
Last modified on  20130322 13:27:38 
Owner  Dr_Absentius (537) 
Last modified by  Dr_Absentius (537) 
Numerical id  6 
Author  Dr_Absentius (537) 
Entry type  Definition 
Classification  msc 55R05 
Synonym  normal covering 
Synonym  Galois covering 