Theorem 19.1:

Let $p : \widetilde X \longrightarrow X$ be a covering projection and $\phi$ be a deck transformation. Then

(i) $\phi$ is uniquely determined by its value at one point of $\widetilde X$

(ii) $\phi(\tilde x_0) \in p^{-1}(x_0)$ whenever $\tilde x_0 \in x_0$.

(iii) If $\phi(\tilde x_1) = \tilde x_2$, where $\tilde x_1, \tilde x_2 \in p^{-1}(x_0)$ then

$$p_{*}\pi_1(\widetilde X, \tilde x_1) = p_{*}\pi_1(\widetilde X, \tilde x_2) \eqno(19.1)$$

(iv) Conversely if (1) holds then there exists a unique deck transformation $\phi$ such that $\phi(\tilde x_1) = \tilde x_2$