I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it?
Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a free action of a finite group $G$ by automorphisms. For all $x,y\in M$, there exists a path $p : [0,1] \to M$ with $p(0)=x$, $p(1)=y$ and such that the image of $p$ intersects each of its $G$-translates transversally.
The corollary I am actually interested in is that if $d\ge 3$ and $x,y$ are not in the same $G$-orbit, you can make the $G$-translates of $p$ pairwise disjoint. If someone has a direct reference for the corollary, that would be even better.
