Find the group of Deck transformation? I am solving questions on Deck Transformation. This is from Munkres.
My efforts 2(a)]
I have proved covering space of graph is again a graph. I have shown that p is a covering map. Since it is 2 sheeted covering space it is regular. Group of Deck transformation is nothing but isomorphic to cyclic group of order $2$.
My efforts for 2(b)
I have proved the following fact. If we have a k-sheeted covering space then size of $Deck(p)$ is less than or equal to $k$.
Since it is a $3$ sheeted covering space, cardinality of $Deck(p)\leq 3$. 
What is the group of Deck Transformation in this case?
How do I show that the covering is normal?

I also need hint for 2(c) and 2(d).


 A: Here are some hints.


*

*The order of the deck group is a divisor of the degree of the cover.  (This statement assumes the deck group is a finite group and the covering is a finite sheeted cover, which is the case in these examples.)

*If a covering $p:(E,e_0) \to (B,b_0)$ is regular (also called normal or Galois in some texts), then a based loop in $(B,b_0)$ that lifts to a based loop in $(E,e_0)$ will also lift to a based loop in $(E,e_1)$ for any choice of basepoint $e_1 \in p^{-1}(b_0)$.  See if this is the case in your examples.
Now, it's important to understand why #1 and #2 are true.  You can probably find the details in Munkres.  The theorem (which is a challenging but worthwhile exercise to prove) that I find especially helpful (and will prove #1) is the following:
the deck group $D$ is canonically isomorphic to $N/K$, where $K = \pi_1(E,e_0)$ and $N$ is the normalizer in $G=\pi_(B,b_0)$ of the image of $K$ in $G$, i.e. of $p_*K$.  Said in fancier language (and making the "canonical" statement precise), there is a short exact sequence
$$1 \to K \to N \to D \to 1,$$ where the map $N \to D$ is given by path lifting.  By "given by path lifting," I mean that if $[\alpha] \in N$ is the homotopy class of an oriented based loop, then we can lift $\alpha$ at $e_0$ to a path with endpoint $e_1$.  There is a unique lift $f:(E,e_0) \to (E,e_1)$ of $p$.  (Existence of this lift uses the fact that $[\alpha]$ belongs to the normalizer.)
You can find a brisk but enlightening treatment of covering spaces using this short exact sequence as the principal tool for computation in the book Algebraic Topology: A First Course by Greenberg and Harper.
