Let $p: X \rightarrow Y$ be a connected covering map. Define the notion of a deck transformation (also known as covering transformation) for $p$. What does it mean for $p$ to be a regular (normal) covering map?

If $p^{-1}(y)$ contains $n$ points for each $y \in Y$, we say $p$ is $n$-to-1. Show that $p$ is regular under either of the following hypotheses:

(1) $p$ is 2-to-1,

(2) $\pi_{1}(Y)$ is abelian.

Give an example of a 3 -to-1 cover of $S^{1} \vee S^{1}$ which is regular, and one which is not regular.