Consider the following sequence of assertions, each of which implies the next. Nothing below has any topology, we just have sets and discrete groups.
- If $F \hookrightarrow E \twoheadrightarrow B$ is fibre bundle of sets (this just means every fibre has the same cardinality), then $|E| = |F \times B|$
- If $H \hookrightarrow E \twoheadrightarrow B$ is a principal bundle, then $|E| = |H \times B|$
- If $H$ is a subgroup of $G$, then $|G| = |H \times G/H|$
- If $H$ is a normal subgroup of $G$, then $|G| = |H \times G/H$|
Question: Do all these assertions fail without the axiom of choice? Or, which of them hold? I actually only care about (3), but for some reason I thought adding these extra statements might somehow clarify things.