For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup

Given: The objects, hypotheses, and choice principles stated above.

Fix a covering $p:E\to B$, a basepoint $b_0\in B$, and $e\in p^{-1}(b_0)$. For $[\alpha ]\in {\pi }_1(B,b_0)$, define $e\cdot [\alpha ]$ as the endpoint of the unique lift of $\alpha$ beginning at $e$ (thm-path-lifting-for-covering-maps). Endpoint homotopy invariance makes this well defined (cor-lifted-path-endpoints-depend-only-on-path-homotopy). With the library's traversal-order product this is a right action; the corresponding left action is $[\alpha ]\cdot e:=e\cdot [\alpha {]}^{-1}$ (def-group-action). (The monodromy right action on a covering fibre and its equivalent left-action convention).

For a covering $p:(E,e_0)\to (B,b_0)$, the induced homomorphism $p_{\ast }:{\pi }_1(E,e_0)\to {\pi }_1(B,b_0)$ is injective. (A covering map induces an injective homomorphism on fundamental groups).

Monodromy acts on each covering fibre by bijections. Its orbit through $e$ is exactly the intersection of the path component of $e$ with that fibre. (Monodromy acts by fibre bijections, and its orbits are the intersections of path components with the fibre).

Let $G$ be a group and let $H\le G$ be a subgroup (def-group, def-subgroup). For $g\in G$, the left coset and right coset of $H$ represented by $g$ are $gH:=\{gh:h\in H\},Hg:=\{hg:h\in H\}\mathrm{.}$ The element $g$ is a representative of these cosets. The notation denotes subsets of $G$; it does not assert that either subset is a subgroup. (Left and right cosets $gH$ and $Hg$ of a subgroup).

Let $H\le G$. The left coset set is $G\mathrm{/}H:=\{gH:g\in G\}\mathrm{.}$ By lem-coset-partition, its elements are exactly the blocks of the coset partition of $G$. The index of $H$ in $G$ is $[G:H]:=\mathrm{\mid }G\mathrm{/}H\mathrm{\mid }$ when $G\mathrm{/}H$ is finite, with finite cardinality as in def-finite-cardinality. If $G\mathrm{/}H$ is not finite, write $[G:H]=\mathrm{\infty }$. Here $\mathrm{\infty }$ is a symbol, not a natural number, and no arithmetic with it is defined. The right coset set has the same finite or infinite size, because lem-left-and-right-cosets-equinumerous gives an explicit bijection between the two coset sets; thus the index does not depend on choosing left rather than right cosets. (The coset set $G\mathrm{/}H$ and the index $[G:H]$ of a subgroup).