Let $G$ a topological group and $H$ a subgroup such that $H$ and $G/H$ are path connected.
Is it true this implies that $G$ is path connected?
I already know that if $H$ and $G/H$ are connected so is G (proposition 1.6.5), but i can't find anything about path connected spaces.
Without further assumptions, this is false. As an example, take the $p$-adic solenoid group $G$: It is the inverse limit of groups $S^1, n\in \mathbb N$ under iterated $p$-fold covering maps. This group is not path-connected, the path-connected component $H$ of the identity in $G$ is isomorphic to $\mathbb R$. The subgroup $H$ is dense in $G$ and, thus, the quotient $G/H$ has trivial topology. Hence, $G/H$ and $H$ are path-connected, but $G$ is not. I do not know what happens if all the groups in question are Hausdorff. If $G/H$ is path-connected and $H$ is closed in $G$ and is a connected Lie group, then $G$ is also path-connected. This is because in this case the projection $G\to G/H$ has the path-lifting property.
