Computing the group of covering transformations of a covering map

Question

Let $B$ be a path connected and locally simply connected space. Let $H$ be a normal subgroup of $\pi_1(B,b).$

If $$p:(E,e) \rightarrow (B,b)$$ is the connected covering map for which $$p_*: \pi_1(E,e) \rightarrow \pi_1(B,b)$$ has image $H$.

Question: How I do compute the group of covering transformations of $p$?

Any help on this will be great.

Answer 1

The fact that the image of $p_*$ is the normal subgroup $H$ implies that your covering map is a regular covering map.

Now there's an important theorem on regular covers that tells that the group of deck transformations (i.e. the group of automorphisms of $p$) is isomorphic to the quotient group $\pi_1(B,b)/p_*(E,e)$.