The mistake you have made is to replace the EMF labelled $2E$ with an open circuit. If you replace it with a short-circuit then the vertical resistance $R$ and the internal resistance $R$ are both shorted out - there is no resistance between F and G.
It is a good idea to re-draw the circuit after shorting the EMFs :
The EMF labelled $E, R$ cannot be shorted because it appears to have an internal resistance of $R$. There is no indication that the EMF labelled $2E$ has any internal resistance. When this is shorted there is no resistance between the points F and G. Points F, G, H are then at the same potential, so the 3 resistors $R$ (horizontal), $2R$ (diagonal) and $2R$ (vertical) are in parallel.
The resistance between A and B is therefore $\frac12 R$.