Theorem 1

Let B be a topological space, let $(E, p)$ be a covering of B, let $b$ be a point of B and let $x$ be a point of the fibre $E_b$.

a) The orbit of $x$ for the canonical operation of the group $\pi_1(B, b)$ is the intersection of $E_b$ and the arcwise connected component of $x$ in E. In particular, if the space E is arcwise connected, this operation is transitive.

b) The stabilizer (A, I, p. 51) of $x$ is the subgroup $p_*(\pi_1(E, x))$ of $\pi_1(B, b)$.

c) For every $\gamma \in \pi_1(B, b)$, we have $p_*(\pi_1(E, x)) =$ Int($\gamma$ )$(p_*(\pi_1(E, x\cdot$ $\gamma )))$.

d) If the space B is connected and the covering E is Galois, the subgroup $p_*(\pi_1(E, x))$ is normal in $\pi_1(B, b)$.

Assertion a) is immediate. Assertion b) follows from Cor. 2 of Prop. 4 (III, p. 303). Assertion c) follows from b) and from Proposition 2 of A, I, p. 52. Finally, suppose that B is connected and that E is a Galois covering of B. For every $\gamma \in \pi_1(B, b)$, there exists a B-automorphism $g$ of E such that $g(x) =x\cdot \gamma ($I, p. 102, th. 2, c)). We have $p=p\circ g$, whence $p_*(\pi_1(E, x)) =p_*(\pi_1(E, x\cdot \gamma ))$. Assertion d) therefore follows from c).