# Non-normal covering of a Klein bottle by torus.

I am trying to construct a non-normal covering of Klein bottle with itself and by a torus. For Klein bottle to Klein bottle, I got a three sheeted covering, just glue three Klein bottles, which is non-normal but whatever covering I am constructing by torus it is coming normal. Any help will be appreciated.

-
Not a single comment yet :( –  ovi Dec 4 '12 at 15:47

Consider the fundamental group of the Klein bottle, $K=\langle x,y\mid y^{-1}xy=x^{-1}\rangle$.
You are looking for a finite index subgroup of $K$ (call it $H$), that is abelian and non-normal. Note that any such covering factors through the orientable double cover, corresponding to the subgroup $\langle x, y^2\rangle$.
Consider $H=\langle xy^{-2}, y^6\rangle$. This is abelian, non-normal, and of index $6$ in $K$. It can be described by the diagram below:
I should mention that $6$ is the smallest number of sheets you can have for a non-normal covering of the torus to the Klein bottle. –  user641 Dec 5 '12 at 4:44