Proposition 5

The mapping $h_{(E,x)}$ of $\pi_1(B, b)$ into G which, to $\gamma \in$ $\pi_1(B, b)$, associates the unique element $g$ of G such that $x\cdot g=x\cdot \gamma^{-1}$, is a group homomorphism, whose kernel is the subgroup $p_*(\pi_1(E, x))$ of $\pi_1(B, b)$. If E is connected, this homomorphism is surjective.

For every $g\in G$, one has $h_{(E,x\cdot g)}=$ Int($g^{-1}$)$\circ h_{(E,x)}$.

Let E be a principal covering of B with group G, let $x$ be a point of $E_b$, and let us denote by $p$ the projection of E. For every $g\in G$, the mapping $y\mapsto y\cdot g$ is a B-automorphism of E. Hence, for every $g\in G$, every $y\in E_b$, and every $\gamma \in \pi_1(B, b)$, one has the relation $(y\cdot g)\cdot \gamma = (y\cdot \gamma )\cdot g($cf. III, p. 305, relation (4)). Consequently, for $\gamma , \delta \in \pi_1(B, b)$, one has

which proves that $h_{(E,x)}$ is a group homomorphism.

Its kernel is the stabilizer of $x$ for the canonical operation of $\pi_1(B, b)$, hence is equal to $p_*(\pi_1(E, x))$ by Theorem 1 of III, p. 305. The mapping $g\mapsto x\cdot g$ is a bijection of G onto $E_b$. The image of the homomorphism $h_{(E,x)}$ is therefore the set of $g\in G$ such that $x\cdot g$ belongs to the orbit of $x$ for this operation. If E is connected, $\pi_1(B, b)$ operates transitively on $E_b($loc. cit.) and the homomorphism $h_{(E,x)}$ is surjective.

Let $g\in G$; for every $\gamma \in \pi_1(B, b)$, one has

whence $h_{(E,x\cdot g)}(\gamma ) =g^{-1}h_{(E,x)}(\gamma )g$. This completes the proof of the proposition.