Corollary 3

Let $(E, p)$ be a connected covering of B and let $x$ be a point of the fibre $E_b$. Let N denote the normalizer of $p_*(\pi_1(E, x))$ in $\pi_1(B, b)$. For every element $\gamma$ of N, there exists a unique B-automorphism $g$ of E such that $g(x) =x\cdot \gamma$, and the mapping $\gamma \mapsto g$ defines by passing to the quotient a group isomorphism of $N/p_*(\pi_1(E, x))$ onto Aut$_B(E)$.

Let $\gamma \in \pi_1(B, b)$; one has $p_*(\pi_1(E, x)) =$ Int($\gamma$ )$(p_*(\pi_1(E, x\cdot \gamma )))$ (III, p. 305, Theorem 1, c)). According to Corollary 2, in order that there should exist a B-automorphism $g$ of E such that $g(x) =x\cdot \gamma$, it is necessary and sufficient that $\gamma$ belong to N. Such an isomorphism is then unique. Let $\alpha : N\rightarrow$ Aut$_B(E)$ denote the mapping $\gamma \mapsto g$ thus defined.

Let $\gamma$ and $\gamma '$ be two elements of N. Put $g=\alpha (\gamma ),g'=\alpha (\gamma ')$; one has $g(g'(x)) =g(x\cdot \gamma ') =g(x)\cdot \gamma '= (x\cdot \gamma )\cdot \gamma '=x\cdot (\gamma \gamma ')$ according to relations (4), III, p. 305 and (1), p. 304. Consequently, $\alpha$ is a group homomorphism. In order that $\alpha (\gamma ) =$ Id$_E$, it is necessary and sufficient that one have $x\cdot \gamma =x$, by virtue of the uniqueness of $\alpha (\gamma )$, that is to say $\gamma \in p_*(\pi_1(E, x))$ (III, p. 305, Theorem 1, b)). Finally, if $g$ is a B-automorphism of E, there exists a path $c$ joining $x$ to $g(x)$ in E (which is arcwise connected), and one then has $g(x) =x\cdot \gamma$, where $\gamma$ is the class of the path $p\circ c$. This proves that the homomorphism $\alpha$ is surjective.

The group Aut$_B(E)$ operates on the fibre $E_b$ and is identified with the automorphism group of the homogeneous (right) $\pi_1(B, b)$-set $E_b($cf. A, I, p. 56, prop. 5 and 6).